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N  DU  STRJ  AL  AR1THMET1 


Southern  Branch 
of  the 

fniversity  of  California 


Los  Angeles 


L  1 


QA 
103 
R69 


This  book  is  DUE  on  the  last  date  stamped  below 


•MY  1  4 


19201 


Z  « 


Form  L-9-15m-8,'26 


INDUSTRIAL  ARITHMETIC 
ROR  AY 


AN  ELEMENTARY  TEXT  FOR  BOYS  IN  INDUSTRIAL, 

TECHNICAL,  VOCATIONAL  AND  TRADES 

SCHOOLS,  BOTH  DAY  AND  EVENING 


BY 

NELSON   L.    RORAY 

DEPARTMENT  OF  MATHEMATICS,   WM.  L.  DICKINSON 
HIGH  SCHOOL,   JERSEY  CITY,  N.  J. 


WITH  86  ILLUSTRATIONS 


PHILADELPHIA 
P.   BLAKISTON'S  SON  &  CO. 

1012  WALNUT  STREET 


COPYRIGHT,  1916,  BY  P.  BLAKISTON'S  SON  &  Co. 


XHK    MAPLK     FKKSS     YUMK    PA 


PREFACE 

The  following  pages  presuppose  a  knowledge  of  the  ordi- 
nary course  of  Grammar  School  Arithmetic. 
They  are  intended: 

First. — To  review  and  give  drill  in  the  mathematical  tools 
needed  by  boys  in  the  shops  during  the  first  year 
of  the  Industrial  High  School. 

Second. — To  give  some  of  the  problems  the  boys  must  handle 
in  the  school  shops  and  may  have  to  handle  in 
practical  life.  Many  of  the  problems  have  been 
taken  from  the  shops  of  the  Wm.  L.  Dickinson 
High  School. 

Third. — To  introduce  the  idea  of  general  positive  number, 
its  use  in  formulae  and  in  simple  equations,  thereby, 
incidentally,  giving  some  preparation  for  the  course 
in  algebra.  No  formal  approach  to  algebra, 
however,  is  intended. 

Fourth. — To  give  the  boy  who  leaves  school  through  neces- 
sity during  the  first  year  of  his  Industrial  High 
School  course  some  of  the  practical  applications  of 
the  most  used  geometrical  formulae. 

The  manuscript  of  this  book  has  been  used  with  about  thirty 
different  classes  under  several  teachers  in  the  Industrial 
Department  of  the  Dickinson  High  School  during  the  past  five 
years.  In  order  that  the  problems  will  be  expressed  in  the 
language  and  data  of  the  shops  and  also  that  they  will  satisfy 
the  actual  mathematical  needs  of  the  first  year  shop  work, 
consultations  were  held  from  time  to  time  with  the  shop 
teachers. 


VI  PREFACE 

The  drawings  were  made  under  the  supervision  of  Mr. 
Stewart  by  the  pupils  of  the  Industrial  Department,  most  of 
them  by  Mr.  Rossback  of  the  Junior  Class. 

The  author  especially  acknowledges  his  indebtedness  to  his 
colleagues  Messrs.  Burghardt,  Steele,  Stewart,  Loomis  and 
Wagner  for  valuable  suggestions  and  assistance  and  also  to  Mr. 
Mathewson  for  his  encouragement  and  help  in  the  preparation 
of  this  book. 

THE  AUTHOR. 

DICKINSON  HIGH  SCHOOL 
JERSEY  CITY,  N.  J. 


CONTENTS 

PAGE. 

LESSONS           I-IV.  Reviews 1-7 

LESSON                  V.  General  Number 8 

LESSON                 VI.  Formulae n 

LESSON               VII.  Angles  and  Polygons 13 

LESSON             VIII.  Measurement,  Woodworking.    .' 18 

LESSON                IX.  Measurement  Drawing 20 

LESSON                  X.  Screw  Threads 23 

LESSON                XI.  Machine  Shop  Measurements 25 

LESSON              XII.  Decimal  Equivalents ' .  27 

LESSON             XIII.  Review 29 

LESSON             XIV.  Area  of  Rectangle 31 

LESSON              XV.  Review 33 

LESSON             XVI.  Square  Root < 35 

LESSON           XVII.  Review 37 

LESSON          XVIII.  Fractional  Review 38 

LESSON             XIX.  Area  of  Parallelograms 39 

LESSON               XX.  Area  of  Triangles 40 

LESSONS  XXI-XXII.  Reviews 42-44 

LESSON         XXIII.  Circles — Area  and  Circumference 46 

LESSON          XXIV.  Speed 48 

LESSON            XXV.  Speeds  of  Pulleys  and  Gears 50 

LESSON          XXVI.  Cutting  Speed  and  Feed .    .    . 54 

LESSON        XXVII.  Area  of  Cylinder 56 

LESSON      XXVIII.  Review 57 

LESSON         XXIX.  Volume  of  Prism 59 

LESSON           XXX.  Review 62 

LESSON         XXXI.  Review 63 

LESSON       XXXII.  Review  of  Percentage 65 

LESSON      XXXIII.  Board  Measure 66 

LESSON      XXXIV.  Review 67 

LESSON        XXXV.  Woodworking  Problems 68 

LESSON      XXXVI.  Volume  of  Cylinder 71 

LESSON     XXXVII.  Review 72 

vii 


Vlll 


CONTENTS 


PAGE. 

LESSON  XXXVIII.  Review 73 

LESSON      XXXIX.  Forge  Shop  Problems 74 

LESSON  XL.  Review 76 

LESSON  XLI.  Simple  Equations 78 

LESSON  XLII.  Theorem  of  Pythagoras 79 

LESSON          XLIII.  Review 81 

LESSON          XLIV.  Factoring  •    •    •_ 83 

LESSON  XLV.  ^/ab  =  \/a.  Vb 84 

LESSON          XLVI.  Review 85 

LESSON         XLVII.  Triangles 86 

LESSON       XL VIII.  30°  Right  Triangle 88 

LESSON          XLIX.  Review 89 

LESSON  L.  60°  Right  Triangle 91 

LESSON  LI.  Review 92 

LESSON  LII.  Equilateral  Triangle,  Altitude  and  Area  .    .     94 

LESSON  LIU.  Regular  Hexagon 95 

LESSON  LIV.  Screw  Threads 97 

LESSON  LV.  Review  .....: 99 

LESSON  LVI.  Thread  Cutting 102 

LESSONS  LVII,  LVIII.  Tapers  and  Taper  Turning  ....    110-113 

LESSON  LIX.  Review 114 

LESSON  LX.  Ratio 116 

LESSON  LXI.  Sectors  and  Segments •  ...   119 

LESSONS  LXII,  LXIII,  LXIV.  Reviews 120-124 

LESSON  LXV.  Area  of  Pyramids  and  Cones 125 

LESSON  LXVI.  Volume  of  Pyramids  and  Cones 127 

LESSONS  LXVII-LXXII.  Reviews 129-139 

LESSON  LXXIII,  LXXIV.  Alloys 140-144 

LESSON  LXXV-LXXVIII.  Print  Shop 145-153 

LESSON       LXXIX.  Fractional  Review 154 


INDUSTRIAL  ARITHMETIC 

LESSON  I 

The  expression  52  means  5X5;  the  expression  34  means 
3X3X3X3.  In  the  expression  52,  5  is  called  the  base  and 
2  the  exponent.  The  exponent  shows  how  many  times  the 
base  is  to  be  taken  as  a  factor,  that  is  multiplied  by  itself. 

1.  Name  the  base  and  the  exponent  of  each  of  the  following 
expressions,  also  state  the  meaning  of  each: 

32;  43;  54:  65;  io2;  io3;  io4;  84;  75. 

2.  Find  the  value  of  each  of  the  above  expressions. 

3.  Expand  each  of  the  following: 
io2;  io3;  io4;  io5;  io6. 

4.  Note  the  number  of  zeros  in  the  expansion  of  each  expres- 
sion in  problem  3.     How  many  zeros  are  in  the  expansion  of 
any  power  of  io? 

5.  State  the  power  of  io  of  each  of  the  following: 
1000;  100;  1,000,000;  100,000,000;  10,000;  io;  100,000. 

6.  Multiply  each  of  the  following  by  io;  by  100;  by  1000: 
24;  24.3;  2.43;  3.1415. 

7.  Compare  the  position  of  the  decimal  point  in  the  product 
with  its  position  in  the  multiplicand  in  each  example  of  problem 
6.     What  effect  upon  the  position  of  the  decimal  point  of  a 
number  has  multiplying  it  by  io?     By  100?     By  io4?     By 
io8?     State  the  principle  for  multiplying  quickly  any  number 
by  io  or  any  power  of  io? 

8.  Write  down  the  value  of  each  of  the  following: 

.0323  X  io3;  3.141592  X  io7;  .000323  X  io2;  32.30  X  io4; 
3I4I-592XI08; 


2  INDUSTRIAL   ARITHMETIC 

323,000  X  io5;  1.732  X  io6;  .01414  X  io2;  .507984  X  io3; 
52  X  io8;  .00005  X  io6;  8976  X  io5;  .001  X  io4;  5.00004  X 
io6;  8.976  X  io4. 

9.  State  a  quick,  easy  way  for  multiplying  a  number  by  20; 
by  300;  by  4000;  by  50,000. 

10.  State  the  result  of  multiplying  each  of  the  following 
numbers  by  20;  by  400;  by  5000;  by  6000: 

123;  3.24;  .0122;  83.016;  12,140. 

11.  Since  25  =  100/4  then  12  X  25  =  12  X  100/4  =  1200/4 
=  300. 

That  is  to  multiply  a  number  by  25  we  move  the  decimal 
point  two  places  to  the  right  and  divide  the  result  by  4. 

12.  Since  33%  =  100/3,  what  is  the  principle  for  multiplying 
a  number  by  33^? 

13.  State  the  principle  for  multiplying  a  number  by  12^; 
by  16%;  by  50;  by  66%;  by  n^. 

14.  Use  the  above  principles  and  multiply  each  of  the  fol- 
lowing numbers  by  25;  by  33^;  by  16%;  by  12^;  by  n^; 
by  50;  and  by  66%: 

17.28;  1.448;  981;  1892;  1658;  3.1416;  2.0524;  78.6;  4.32; 
1214.0678;  3.572. 

16.  At  12^^.  per  ft.  what  will  1448  ft.  of  concrete  cost?  At 
i6%?<.  per  ft.?  At  n}^.  per  ft.? 

16.  How  many  feet  will  a  line  12  in.  long  represent  if  the 
scale  is  i  in.  =  12^  ft.?    If  i  in.  =  33)^  ft.?    If  i  in.  = 
25  ^.? 

17.  If  a  cubic  foot  of  iron  weighs  480  Ib.  how  many  pounds 
will  a  plate  weigh  that  contains  16%  cu.  ft.?     25  cu.  ft.? 
12^  cu.  ft?    66%  cu.  ft.?    75  cu.  ft?     n>^  cu.  ft.? 


LESSON  II 
REVIEW 

Divide  175.6  by  10;  by  100;  by  1000. 

Compare  the  position  of  the  decimal  point  of  the  quotient 
with  that  of  the  dividend. 

State  a  short  easy  method  for  dividing  a  number  by  10  or 
any  power  of  10. 

1.  State  the  quotient: 

56.34  -T-  10;  76.54  -r-  1000;  587  -r-  100;  .057  -f-  100;  563.4 
-j-  100;  893.2  -7-  10;  68.3  -r  1000;  .003  -r-  10;  5.634  -f-  1000; 
8932  -T-  10,000;  .78  -T-  10,000;  .056  -f-  1000;  .789  4-  10;  .0683 
-f-  100;  .06  -T-  100;  .56  -r-  10. 

2.  State  a  short  easy  method  for  dividing  a  number  by  20; 
by  300;    by  4000;  by  3000;  by  80. 

3.  State  the  quotient: 

248  -H  20;  67.5  -T-  500;  7593  -f-  5000;  89  -T-  400;  248  -T-  40; 

7-86  -7-   200;   759.3   -4-   5000;   8.9   -T-  40;   246   -T-  600;  3675   -7- 
2000;  7.593  -T-  500O;  .89  -5-  4000;  1728  -7-  1200;  8958  -7-  5000; 

•7593  •*•  5°°°;  76  •*•  J900- 

4.  At  90 j£.  per  1000  cu.  ft.  what  will  2486  cu.  ft.  of  gas  cost? 

5.  How  many  tons  in  5240  lb.?     In  137,801  lb.?     In  756 
lb.? 

6.  At  $25  per  1000  what  will  12,280  bd.  ft.  of  lumber  cost? 

7.  At  $12^  per  1000  find  the  cost  of  16,300  bricks. 

8.  At  $5  per  ton,  how  much  must  be  paid  for  12,340  lb.  of 
coal? 

9.  A  gallon  of  paint  will  cover  200  sq.  ft.  of  surface  two  coats. 
State  a  rule  for  finding  the  number  of  gallons  of  paint  required 
to  give  any  surface  two  coats. 

3 


INDUSTRIAL    ARITHMETIC 


10.  Find  the  number  of  gallons  of  paint  required  for  two 
coats  for  2578  sq.  ft.;  for  3780  sq.  ft.;  for  160  sq.  ft. 

11.  A  quart  of  grass  seed  is  sufficient  for  300  sq.  ft.  of  lawn. 
How  many  quarts  will  be  required  to  seed  2970  sq.  ft.? 


LESSON  III 
REVIEW 

100 

Since  2<  = then 

4 
100        1200  X  4 

1 200  -T-  2<;  =  1 200  -s —  -  =  -  -  =  48. 

4  100 

That  is,  to  divide  a  number  by  2$,  move  the  decimal  two  places 
to  the  left  and  multiply  result  by  4. 

1.  Divide  each  of  the  following  by  25: 
1660;  72.86;  56;  56.4;  1728;  3564;  3.1416. 

2.  Study  the  above  and  find  a  short,  easy  way  for  dividing 
a  number  by  33^;  50;  16%;  11%;  14^;  66%. 

3.  State  the  quotient: 

1800  -T-  33^;   12.34  -4-  12}^;  35.81  -f-  50;  1234  -i-  50;  27.16 

•*-  33^5  35-7  *•  "M;  17-60  -5-  14^;  75-*6  -s-  25;  87  -i-  33^; 
3.46  -T-  16%;  18.32  ^-  n^;  1765  -T-  33^;  128  4-  33^;  2468 
*  16%;  175-3  -s-  66%;  18.32  -i-  14^. 

4.  In  problem  3  change  each  division  sign  to  a  multiplica- 
tion sign  and  state  product. 

6.  It  is  required  to  cut  a  i6-ft.  bar  of  steel  into  pieces 
12^6  m-  long;  how  many  pieces  can  be  cut  from  the  bar, 
allowing  Ke-in.  waste  for  each  piece  in  cutting? 

6.  At  i6%j£.  per  Ib.  how  many  pounds  of  copper  wire  can  be 
bought  for  $18?     For  $2400?     For  $300? 

7.  How  many  taper  shanks  24%  in.  long  can  be  cut  from  a 
round  piece  12  ft.  long,  if  ^  in.  to  each  piece  is  wasted  in 
cutting? 

8.  A  man  bought  a  building  lot  at  $16%  per  ft.     What  did 
he  pay  for  a  68-ft.  lot? 

9.  Find  a  short  easy  way  for  multiplying  a  number   by 
99,  by  101. 

Hint. — 99  =  100  —  i. 

s 


+ 


+ 


=  ? 


+ 


-  H 


54  =   ? 

-H- 


2. 


- 


2 


29%  +  H  -  15  +  H  -  uK  =  ? 


3.  Multiply  15%  by  12;  18%  by  8;  17%  by  8;  28^  by  8. 

4-  %  x  7<f  x  Ho  =  ?  H2  x  y±  x  %  =  ? 

I/  \x  I/  \/  IX  3  5X  v   16X     \x  45X        -   ? 

72  ^>  72  />  /2  —   •  /8  A     x25  ^     x64  —   ' 

X  %  X  %  =  ?        5^  X  3  X  2  =  ? 
X  2H±  X%=  ?          2%  X  5  X  8  =  ? 

6.  Divide  !%4  by  2;  3^  by  37<f;  28  by  3^;  3^  by  2%; 
)  by  %. 

16  -4-  sM  =  ?  18-^  i67^  =  ? 


KG  -s-  5  = 


REVIEW  7 

7.  From  15^  X  3  take  5^  X  4  and  divide  the  difference 


8.  Multiply  3.1416  by  26;  by  2.6;  by  .26;  by  .026. 

9.  Divide  each  of  the  following  by  3.1416  and  carry  the 
division  to  two  decimal  places: 

15;  23.48;  178.532;  17;  2;  4-8- 

10.  Multiply  .015625  by  i2>^;  by  25;  by  16%; 

11.  Change  to  decimal  fractions: 


*-/n        *-/A        *-/a        *•/*  n        a/o  n        *  "ri?  ^ 

72  >   74  >    78  >    716)    73  2  >      764- 

12.  Change  to  common  fractions  in  their  lowest  terms: 
.15;  .015625;  .0625;  .125;  .03125;  .235. 


LESSON  V 
GENERAL  NUMBER 

In  our  previous  work  all  the  numbers  we  have  used  have  had 
particular  values  and  were  represented  by  a  definite  symbol. 
For  example  the  symbol  5  stands  for  a  group  of  five  units. 
We  shall  now  use  other  symbols  to  represent  numbers  which 
may  have  any  values  whatever,  or  numbers  whose  values  are,  as 
yet,  unknown,  e.g.,  we  speak  of  a  rod  a  ft.  in  length  meaning 
any  number  of  feet,  of  x  marbles  meaning  any  number  of 
marbles,  etc. 

1.  If  John  has  $6  and  Henry  has  $5,  together  they  will  have 
$6  +  $5. 

2.  If  John  has  $a  and  Henry  has  $b,  together  they  will  have 
$a  +  $b. 

3.  In  which  of  the  above  problems  are  numbers  used  that 
are  represented  by  particular  values?     That  are  represented 
by  any  value  whatever? 

4.  If  in  problem  2,  a  stands  for  10  and  b  stands  for  5,  how 
many  dollars  have  they  together?    If  a  =  7  and  b  =  8?     If 
a  =  12  and  b  =  10?    If  a  =  13  and  b  =  17? 

5.  Read  each  of  the  following  and  tell  what  operation  is  in- 
cluded: 

(a)  7 +  5;    8X3;    9-*- 35    8-7J    9  +  55    5  +  35  9-6; 
7  X  55  16  -^  4;  7  -  5;  8  -^  7;  8  +  10. 

o 

(b)  a  +  b;  a  —  b;  a  X  b;  a  •*•  b;  t;  3a;  3b;  4c;  2a  +  3c; 

b 

4x  X  3bJ  3  X  n;  5  -=-  k;  |;  7ab;  gef;  (x  -  i)  -4-  p;  a 

X  c  +  d;  n  —  5;  6  +  8b;  3a  —  sb  +  cd. 
8 


GENERAL   NUMBER  9 

Express  with  the  proper  symbol  of  operation  the  solution 
of  each  of  the  following: 

6.  Tom  has  10  marbles  and  Frank  has  7  marbles,  how  many 
marbles  have  the  boys  together?     If  Tom  has  a  marbles  and 
Frank  has  r  marbles? 

7.  What  is  the  perimeter  of  a  square  if  a  side  is  6  ft.?     S  ft.? 
Tft?    Xft.?     Bft.?     10  ft.? 

8.  By  Jiow  much  does  n  exceed  8?     12  exceed  9?     7  exceed 
5?     10  exceed  k?    k  exceed  3?    e  exceed  f?    r  exceed  t? 
y  exceed  2p? 

9.  If  the  age  of  a  boy  is  now  16  years,  how  old  was  he  7 
years  ago?     8  years  ago?    What  operation  is  used  to  solve 
this  problem? 

How  old  was  he  a  years  ago?    d  years  ago?    q  years  ago? 

10.  At  $3  each  how  many  hats  can  be  bought  for  $18?    For 
$24?     For  $36?     For  $a?     For  $f?     For  $h? 

11.  At  $a  each  how  many  books  can  be  bought  for  $15?  For 
$13?     For  $b?     For  $v?    For  $x?     For  $i? 

12.  At  $2  each  what  will  6  hammers  cost?  8  hammers?     a 
hammers?     c  hammers?     k  hammers? 

13.  How  many  feet  longer  is  a  i2-ft.  stick  than  a  lo-ft. 
stick?     A  i5-ft.  stick  than  a  i2-ft.  stick?     An  a-ft.  stick  than 
a  b-ft.  stick? 

Make  a  word  problem  for  each  of  the  following: 

14.  7  +  5;  a  .+  b;  a  -  b;  $a  X  b;  $a  •*-  $G;  $a  4-  $y; 

3a;  Sb- 

15.  What  is  the  sum  of  3  ft.  and  5  ft.?     Of  3a  and  $a? 
Of  7a  and  6a?  Of  3m  and  5m?  Of  3a  and  5a? 

16.  What  is  the  difference  between  8  books  and  5  books? 
Between  8x  and  3X?     Between  10!  and  3!?     Between  8s  and 
53?     Between  7r  and  $r? 

17.  sa  +  6a  -  2a  +  4a  =  (  )a;  izy  -  8y  +  icy  -  $y  =  ?; 
3b  +  7b  -  5b  +  8b  =  (  )b;  3k  +  4k  -  5k  -  2k  =  ?;    4*  + 
5X  —  8x  +  iox  =  ?;  8s  +  75  —  IDS  —  55  =  ?;    8^4!  +  2^1  — 


10  INDUSTRIAL   ARITHMETIC 

2^1  =  ?    3m  +  iom  -  8m  -  2m  = 


18.  Find  the  value  of  each  of  the  following  expressions  if 
a  =  i,  b  =  2,  c  =  3: 

a  +  b;  c  —  d;  c  +  b  —  a;  (20  +  b)  -J-  2;  (8a  —  2)  -f-  b; 
c  +  b  +  a;  aXbXc;  b  X  c  —  a;  2c  ^  b  -f-  5;  40  + 


5  - 

2       be       be       b 

19.  The  expressions  ab,  a-b,  a  X  b  all  mean  that  a  is  to  be 
multiplied  by  b. 

The  expression  a(b  +  c  -f-  d)  means  that  b,  c  and  d  are  to 
be  added  and  the  sum  multiplied  by  a.  When  2  or  more  num- 
bers have  no  sign  of  operation  between  them  like  abed,  it  is 
to  be  understood  that  the  numbers  are  to  be  multiplied. 


LESSON  VI 
FORMULAE 

The  statement  of  a  rule  by  means  of  general  numbers  and 
other  mathematical  symbols  is  called  a  formula.  For  exam- 
ple, the  rule  for  finding  the  area  of  a  circle  is,  multiply  the 
square  of  the  radius  by  3.1416.  This  rule  stated  by  means  of  a 
formula,  if  S  =  area  of  circle,  T  =  3.1416  and  r  radius  of  the 
circle,  is  S  =  Trr2. 

o  p 

1.  If  r  and  -T  be  any  two  fractions,  state  by  means  of  a 

b         d 

formula  the  rule  for  the  multiplication  of  two  fractions;  for 
the  division. 

2.  State  as  a  formula  the  following:  The  area  of  a  rectangle 
is  equal  to  its  base  multiplied  by  its  altitude.    Let  b  =  base 
and  h  =  altitude. 

3.  If  b  =  the  area  of  the  base,  h  =  the  altitude  and  v  = 
the  volume  of  a  cone,  from  v  =  ^bh,  state  in  words  the  rule 
for  finding  the  volume  of  a  cone. 

4.  The  formula  for  finding  volume  (v)  of  a  sphere  whose 
radius  is  r  is  v  =  ^jTrr3.     Find  volume  of  a  sphere  whose  radius 
is  5,  6,  8,  10. 

6.  L  =  length  of  open  belt  in  feet  (approximate).  D  = 
distance  between  centers  of  pulleys  in  feet.  R  and  r  =  radii 
of  two  pulleys  in  feet. 

r)  +  2d. 


The  distance  between  the  centers  of  two  pulleys  is  20  ft.  and 
the  radii  of  the  pulleys  18  in.  and  n  in.  Find  the  approximate 
length  of  the_open  belt  required  for  the  pulleys. 


12  INDUSTRIAL   ARITHMETIC 

6.  The  approximate  distance  a  body  will  fall  from  rest  in  any 
number  of  seconds  is  given  by  D  =  i6t2. 

Find  the  distance  a  body  will  fall  from  rest  in  4  sec.;  10 
sec.;  15  sec.;  i  min. 

7.  C  =  Circumference  of  a  circle  and  r  =  its  radius. 

C  =  27rr. 

What  is  the  velocity  of  a  point  on  the  rim  of  a  wheel,  radius  3  ft. 
making  12^  revolutions  per  second. 


LESSON  VII 
ANGLES  AND  POLYGONS 

1.  The  figure  BAG  is  called  the  angle  BAG.  The  point  A  is 
called  the  vertex  of  the  angle  and  the  straights  AB  and  AC  are 
called  the  sides  of  the  angle.  In  reading  an  angle  the  vertex  is 
always  read  between  the  other  two  letters,  as  angle  BAG;  writ- 


FIG.  i. 


FIG.  2. 


ten  ZBAC.     An  angle  is  often  named  from  its  vertex  letter 
only,  as  the  above  is  called  the  angle  A;  written  Z  A. 

2.  Read  the  angles  of  Fig.  2. 
These  angles  may  be  called  Zi,  Z2,  ^3,  etc. 


Right  Angle 


FIG.  3. 


0 
FIG.  4. 


3.  The  angles  a  and  b  are  called  vertical  angles.    Name 
another  pair  of  vertical  angles  in  the  above  figure.     Name  pairs 
of  vertical  angles  in  the  figure  for  example  2. 

4.  The  angles  a  and  d  are  called  adjacent  angles.     Name 
other  pairs  of  adjacent  angles  in  the  above  figures. 

13 


14  INDUSTRIAL  ARITHMETIC 

6.  Two  angles  are  equal  if  their  sides  can  be  made  to  coincide. 

6.  If  two  straights  so  intersect  that  any  pair  of  adjacent 
angles  formed  are  equal,  the  straights  are  said  to  be  perpen- 
dicular to  each  other  and  the  angles  formed  are  right  angles. 

If  the  ZABE  =  ZDBA  (Fig.  4),  then  the  straight  AC  is 
perpendicular  to  the  straight  DE  and  the  angles  formed  are 
right  angles.  Name  the  right  angles  in  Fig.  4. 

7.  Right  angles  and  perpendiculars  are  often  constructed  by 
means  of  the  T  square  or  a  right  triangle. 

8.  An  angle  is  often  measured  by  the  number  of  degrees  it 
contains.     An  angle  of  one  degree  (i°)  is  one  of  the  90  equal 
angles  into  which  the  right  angle  can  be  divided. 

How  many  degrees  in  one  right  angle?  Two?  Three? 
Four?  %  right  angle?  >£?  i}£?  %?  ^? 

A  protractor  is  used  to  measure  an  angle  in  degrees. 

9.  If  the  sides  of  the  angle  lie  in  the  same  straight  the  angle 
is  called  a  straight  angle,  e.g.,  ZDBE  is  a  straight  angle  (Fig.  4). 

Name  other  straight  angles. 

How  many  degrees  in  a  straight  angle?  How  many  right 
angles? 


Obtuse 


B 
FIG.  5-  FlG-  6- 

10.  An  angle  less  than  a  right  angle  is  called  an  acute 
angle. 

11.  An  angle  greater  than  a  right  angle  but  less  than  a 
straight  angle  is  called  an  obtuse  angle. 

12.  Straights  in  the  same  plane  that  have  no  common  point 
are  called  parallel  lines. 


ANGLES   AND   POLYGONS  15 

Parallel  lines  are  constructed  by  means  of  the  parallel  ruler 
or  the  T  square. 
13.  A  curved  line  is  a  line  no  part  of  which  is  straight. 


Parallels 
FIG.  7.  FIG.  8. 

14.  A  line  not  straight  but  no  part  of  which  is  curved  is 
called  a  broken  line  (Fig.  8). 

B 


ED  DC 

Polygon  Parallelogram 

FIG.  9.  FIG.  io. 

16.  If  the  end  points  of  a  broken  line  coincide  the  figure 
formed  is  called  a  polygon. 

AB,  BC,  CD,  etc.,  are  the  sides  of  the  polygon.  The  angles 
FAB,  ABC,  etc.,  are  the  angles  of  the  polygon. 

The  broken  line  is  called  the  perimeter  of  the  polygon. 

Exercise. — If  each  side  of  the  above  polygon  is  7  in.  what  is 
the  length  of  its  perimeter?  If  5  in.?  If  io  in.?  If  3  ft.? 
If  5  ft.  4  in.? 

16.  A  polygon  of  three  sides  is  a  triangle.     A  polygon  of 
four  sides  is  a  quadrilateral. 

17.  If  the  opposite  sides  of  a  quadrilateral  are  parallel  the 
figure  is  called  a  parallelogram. 

Exercise. — How  many  sides  has  a  parallelogram?    Why? 
Can  the  sides  of  a  parallelogram  be  curved?     Why? 

18.  Facts  relating  to  a  parallelogram:    Draw  a  parallelo- 


i6 


INDUSTRIAL   ARITHMETIC 


gram  and  letter  it  A,  B,  C,  D.  Use  the  compasses  and  com- 
pare AB  with  CD.  Are  they  equal? 

Compare  AD  with  BC.     Are  they  equal? 

Draw  another  and  make  the  same  comparison. 

Compare  the  opposite  angles  in  each  parallelogram. 

The  above  exercises  illustrate  the  following  principles: 

1.  The  opposite  sides  of  a  parallelogram  are  equal. 

2.  The  opposite  angles  of  a  parallelogram  are  equal. 
Exercise. — If  in  the  parallelogram  ABCD,  AB  =  10  in., 

CD  =  ?  Why?  AD  =  3  in.,  BC  =  ?  Why?  CD  =  15  ft., 
AB  =  ?  Why?  BC  =  3  ft.  4  in.,  AD  =  ?  Why?  B  =  78°, 
D  =  ?  C  =  102°,  A  =  ? 

THE  RECTANGLE 

A  B 


C  D 

Rectangle 

FIG.  ii. 

19.  A  parallelogram  having  one  of  its  angles  a  right  angle  is 
called  a  rectangle. 

Exercise. — Are  the  opposite  sides  of  a  rectangle  equal? 
Why?  Are  the  opposite  angles  of  a  rectangle  equal?  Why? 

A         


FIG.  12. 

At  least  how  many  of  the  angles  of  a  rectangle  are  right 
angles? 

Find  by  means  of  a  right  angle  whether  the  other  angles  of 
a  rectangle  are  right  angles. 


ANGLES   AND   POLYGONS  17 

20.  Any  one  of  the  sides  of  a  parallelogram  is  called  its 
base;  e.g.,  AB,  or  BC  or  AD  or  CD  is  a  base  of  the  parallelo- 
gram. 

Name  the  bases  of  the  rectangle  in  example  19. 

21.  The  perpendicular  from  one  side  of  a  parallelogram  to 
the  opposite  side  is  the  altitude  of  the  parallelogram,  e.g.,  AF 
in  the  figure  for  example  20. 

Name  the  altitude  of  the  rectangle  in  example  19  if  DC  is  the 
base;  if  AD  is  the  base. 

THE  SQUARE 


FIG.  13. 

22.  A  rectangle  with  a  pair  of  intersecting  sides  equal  is 
a  square. 

Exercise. — Is  a  square  a  parallelogram?    Why? 

How  many  degrees  in  at  least  one  angle  of  a  square?  Why? 
In  each  angle  of  a  square?  Why?  Are  the  sides  of  a  square 
equal?  Why?  Name  the  bases  and  altitudes  of  the  above 
square.  If  one  base  of  a  square  is  2  in.,  what  is  the  length  of 
its  altitude?  Draw  a  square  one  of  whose  sides  is  12  in., 
letting  i  in.  =  6  in.  Draw  a  rectangle  with  base  20  ft.  and 
altitude  4  ft.  (scale  i"  =  4'  o".)  Draw  a  i-in.  square,  or  i 
sq.  in.;  a  i-ft.  square  or  i  sq.  ft. 


I    LESSON  VIII 
MEASUREMENTS 

Measuring  a  line  consists  in  finding  the  number  of  standard 
units  of  length  it  contains. 

There  are  many  standard  units  of  length  in  common  use, 
such  as  the  inch,  foot,  yard,  etc. 

The  number  of  standard  units  of  length  a  line  contains  is 
the  length  of  the  line  in  terms  of  that  unit;  e.g.,  a  line  is  5  ft. 
long  if  it  contains  the  foot  five  times. 

In  the  wood  working  shops  the  ruler  used  for  measuring 
lengths  is  divided  into  inches  and  each  inch  divided  into  16 
equal  parts,  the  wood  worker  thus  measures  to  ^Q  in.  In  the 
machine  shops  one  of  the  instruments  used  enables  the  machin- 
ist to  measure  to  ;Kooo  m-  This  instrument  is  called  a  microm- 
eter and  is  used  for  measuring  fractional  parts  of  an  inch. 
This  instrument  records  its  measurements  in  the  decimal  scale 
instead  of  in  terms  of  common  fractions.  The  ruler  of  the 
wood  worker  reads  %  in.  whereas  the  micrometer  of  the 
machinist  would  read  .125  in.  for  the  same  length.  The  ordi- 
nary micrometer  registers  accurately  to  .001  in.  The  experi- 
enced machinist  can  measure  very  closely  with  it  to  .0001  in. 

WOOD  WORKING  MEASUREMENTS 

1.  Measure  as  accurately  as  possible  the  top  of  your  desk. 

2.  Find  the  number  of  feet  in  the  length  of  the  schoolroom ; 
the  number  of  yards. 

3.  Measure   the   length   and   width   of   your   schoolroom. 
How  many  feet  of  baseboard  are  required  for  the  room, 

making  deductions  for  all  openings  in  the  baseboard? 

18 


MEASUREMENTS 


4.  Find  the  cost  of  the  chalk  trays  of  the  blackboards  in 
your  schoolroom  at  12%^.  per  ft. 

6.  Determine  the  number  of  feet  of  moulding  in  the  panels 
of  the  doors  of  the  schoolroom  and  its  cost  at  $}^>i.  per  ft. 

6.  How  many  strips  of  floor  moulding  16  ft.  long  will  be 
necessary  for  the  schoolroom? 

7.  How  many  feet  of  picture  moulding  will  be  required  for 
the  room  if  the  moulding  is  at  the  top  of  the  walls?     If  3  ft. 
from  the  top  of  the  walls? 

-5'- 
Floor  Plan 


-20' 


FIG.  14. 


8. 


The  above  is  a  floor  plan. 

(a)  Find  cost  of  the  floor  moulding  for  it  at  8^.  per  ft. 

(b)  Find  cost  of  the  baseboard  at  20^.  per  ft. 

9.  What  is  the  perimeter  of  a  rectangular  room  M  ft.  long 
and  R  ft.  wide?  S  ft.  long  and  12  ft.  wide?  K  ft.  wide  and 
2K  ft.  long? 


LESSON  IX 


DRAWING  MEASUREMENTS 

The  inch  of  the  scale  used  for  measuring  in  the  drawing  room 
is  divided  into  32  equal  parts.  To  what  fraction  of  an  inch 
can  the  draughtsman  measure?  In  the  drawing  room  working 
plans  for  the  machinist,  the  carpenter,  the  pattern  maker, 
etc.,  are  made.  These  drawings  are  seldom  made  full  size 
but  are  drawn  to  a  scale,  that  is,  each  inch  of  the  drawing 
represents  i  ft.  or  2  ft.,  etc.,  of  the  actual  size  of  the  object 
drawn.  When  making  drawings  it  frequently  happens  that 
a  line  must  be  drawn  whose  length  contains  ^2  in-  or  even 
3^4  in.  In  ordinary  drawings  %4  in.  is  neglected,  no  at- 
tempt being  made  to  draw  any  line  shorter  than  ^2  m- 


Scale  K"  =  4'  o" 
FIG.  15. 

Notice  that  the  plan  has  marked  on  it  the  actual  lengths  of 
the  lines  represented,  but  the  lengths  of  the  drawing  are  made 
in  accordance  with  the  scale  selected. 


Exercises 

1.  If  the  scale  is  i"  =  3'  o",  how  long  a  line  must  be 
drawn  to  represent  each  of  the  following: 


DRAWING    MEASUREMENTS 


21 


g: 

in.;  3^  in.;   5.5  in.;  6.25  in.; 


9  ft.  o  in.;  12  ft.  o  in.;  15  ft.  o  in.;  6  ft.  o  in.;  18  ft.  o  in.; 
100  ft.  o  in.;  8  ft.  6  in.;  7  ft.  5  in.;  15  ft.  8  in.;  18  ft.  ^£2  in-J 
25  ft.  K2  in.;  9  in.;  15  in.;  2  in.? 

2.  If  the  scale  is  i"  =  o'.  8",  determine  the  length  of  each 
of  the  following  for  the  drawing: 

i  in.;   %  in.;   %  in.;   2^ 
4.125  in. 

3.  In  the  above  figure  what  should  be  the  length  of  each  line? 

4.  Fill  in  the  following,  the  scale  being  i"  =  3". 

Length  of  line  in  Actual  length  represented 

drawing  by  drawing 

6        in.  ? 

9        in.  ? 

<  2        in.  ? 

in.  ? 

in.  ? 

?  16.5  in. 

5.  Calculate  the  length  of  each  line  on  the  drawing  for  the 
miter  box  if  the  scale  is  half  size;  if  one-quarter  size.     (Fig.  16.) 


FIG.  1 6. 

6.  What  is  the  length  of  each  line  in  the  drawing  for  the 
mortar  box,  if  the  scale  is  W  =  o'.  i"?     (Fig.  17.) 


22 


INDUSTRIAL   ARITHMETIC 


FIG.  17. 

7.  The  scale  for  the  saw  horse  is  3"  =  i'  o".     Calculate 
the  length  of  each  line  for  the  jlra  wing.     (Fig.  18.) 


FIG.  1 8. 

8.  The  side  of  a  filing  cabinet  is  to  be  14  in.  long  10%  in. 
high,  the  stile  and  rail  each  i  ^  in.  wide.  Calculate  the  dimen- 
sions for  a  drawing* with  scale  %"  =  i'  o";  }/±'  =  i'  o". 


LESSON  X 
SCREW  THREADS 

1.  If  a  cylinder  is  spirally  grooved  or  has  a  thread  wound 
spirally  around  it,  a  screw  is  formed.     The  nature  of  a  screw 
depends  upon  the  diameter  of  its  body,  the  number  of  threads 
per  inch,  and  the  shape  of  the  thread. 

2.  Upon  the  number  of  threads  per  inch  depend  the  pitch 
of  the  screw  and  the  lead  of  the  screw.     The  pitch  of  a  screw  is 
the  distance  between  the  top  of  one  thread  and  the  top  of  the 
next.     For  example,  if  a  screw  has  8  threads  per  in.,  the  pitch 
is  evidently  ^  in. 

3.  The  lead  of  a  screw  is  the  distance  it  advances  in  one 
revolution.     In  a  single-thread  screw  with  10  threads  per  in. 
the  lead  is  J^o  in. 

Exercises 
(The  following  problems  refer  to  screws  of  single  thread.) 

1.  A  screw  has  12  threads  per  in.     What  is  its  pitch?     Its 
lead? 

2.  Find  the  pitch  and  lead  for  the  following  number  of 
threads  per  inch : 

3;  4;  5;  6;  8;  n;  20;  13;  40 

3.  A  screw  that  has  40  threads  per  inch  will  advance  how  far 
in  one  complete  revolution  of  the  screw? 

4.  Find  the  number  of  threads  per  inch  for  the  following 
leads: 

K  in.;  %  in.;  %  in.;  ^0  in.;  %  in.;  %  in.;  ^0  in. 

23 


24  INDUSTRIAL   ARITHMETIC 

5.  What  is  the  pitch  of  a  screw  that  has  a  threads  per  in.? 
Its  lead? 

6.  Does  the  lead  of  a  single-thread  screw  equal  its  pitch? 
Why? 

7.  If  the  lead  of  a  screw  is  }/±  in.,  what  distance  will  it 
advance  in  ^  revolution?     Y±  revolution?     }/£5  revolution? 
%  revolution? 

8.  A  screw  has  40  threads  per  in.;  what  distance  will  it 
advance  in   ^5   revolution?   %5?   K?  ^5?  2^5?  !J£5? 

2?    10?    15?    20?    40? 

9.  Express  as  a  decimal,  correct  to  three  places,  the  lead  of 
each  of  the  following  number  of  threads  per  inch: 

40;  20;  10;  8;  12;  7 

10.  Express  each  of  the  results  in  problem  8  as  a  decimal 
correct  to  three  places. 


LESSON  XI 
MACHINE  SHOP  MEASUREMENTS 

The  measuring  instruments  of  the  machine  shop  are  a  6-in. 
scale  graduated  on  one  edge  to  ^4  in.  and  the  micrometer.  The 
6-in.  scale  is  used  for  rough  measurements  and  the  micrometer 
for  more  accurate  measurements.  The  readings  of  the  mi- 
crometer, as  already  stated,  are  in  the  decimal  scale. 

The  micrometer  is  essentially  a  screw  with  40  threads  per 
in.  What  is  its  lead?  How  far  in  the  decimal  scale  will  it 
advance  for  ^5  of  a  revolution?  How  then  is  .001  in. 
measured  with  the  micrometer?  .002  in.?  .006  in.?  .oi5in.? 
.026  in.? 


.Thimble 


FIG.  19. — A  micrometer  screw. 

The  names  of  the  different  parts  of  a  micrometer  are  given 
in  the  figure. 

The  spindle  is  made  to  move  backward  and  forward  within 
the  barrel  by  turning  the  thimble,  to  which  the  spindle  is 
fastened.  The  concealed  end  of  the  spindle  is  a  screw  con- 
taining 40  threads  to  the  inch.  On  the  edge  of  the  thimble  are 
25  divisions  equally  distant.  By  turning  one  division  of  the 
thimble,  the  spindle  is  made  to  move  through  ^5  of  %Q  in-  or 

25 


26  INDUSTRIAL   ARITHMETIC 

.001  in.  Each  revolution  of  the  thimble  is  indicated  on  the 
barrel  by  means  of  a  small  mark  and  every  fourth  revolution 
of  the  thimble  by  a  longer  mark.  The  barrel  is  thus  divided 
for  the  space  of  i  in. 

Exercises 

1.  How  many  divisions  of  the  thimble  must  be  turned  in 
order  to  have  the  micrometer  measure  each  of  the  following: 

.003  in.;  .005  in.;  .010  in.;  .025  in.;  .018  in.;  .075  in.;  .125 
in.? 

2.  What  part  of  an  inch  is  the  distance  between  any  pair  of 
consecutive  small  divisions  of  the  barrel?     Between  any  pair 
of  consecutive  large  divisions  of  the  barrel? 

3.  What  part  of  an  inch  is  the  distance  between  6  consecu- 
tive divisions  of  the  barrel?     8?  10?  15?  2O?4O?4i? 

4.  How  many  revolutions  of  the  thimble  for  each  of  the 
following: 

.025  in.;   .050  in.;   .075  in.;   .150  in.;   .200  in.;  .3  in.;  .5  in.? 

5.  What  decimal  part  of  an  inch  do  the  following  barrel 
readings  represent: 

5  large  divisions;  3  large  divisions;  6  large  divisions;  10 
large  divisions;  4  large  divisions;  4  large  divisions  and  2  small 
divisions;  3  large  divisions  and  3  small  divisions;  2  small 
divisions? 

6.  Express  as  the  decimal  part  of  an  inch  the  following 
readings: 

5  large  and  3  small  of  the  barrel  and  12  of  the  thim- 
ble; 3  large  and  2  small  barrel,  and  16  thimble;  i  large  of  the 
barrel  and  8  thimble;  6  large  and  3  small  barrel,  and  14.5 
thimble;  9.5  thimble. 


LESSON  XII 
DECIMAL  EQUIVALENTS 

As  we  have  already  learned  the  machine  shop's  fractional 
measurements  are  in  multiples  of  ^4  in.;  also,  if  these  meas- 
urements are  to  be  accurate  the  micrometer  is  used.  Hence, 
in  order  to  measure  ^4  in.  its  equivalent  as  a  decimal  must 
be  known.  Why? 

Show  that  ^54  =  .015625.     (Learn  this  result.) 

In  setting  the  micrometer  for  ordinary  work  three  decimal 
places  only  are  considered,  that  is  J^4  =  -OI5-  If  it  is 
required  to  set  the  micrometer  for  3^2  or  %4  we  consider 
3^54  =  .0156,  because  to  get  ^54  we  multiply  ^4  by  2, 
%2  —  %4  =  -031;  that  is  ^34  =  .0156  whenever  it  is  to  be 
multiplied  by  any  number. 

By  means  of  the  following  it  is  possible  to  state  quickly  the 
decimal  equivalent  of  any  number  of  64ths. 


=  •25 
=  -125 

=  .015625 

Express  as  a  decimal  1J^4- 

Solution.—  1%4  =  i%4  +  ^4  =  K  +  ^4  =  -25  +  .015 
=  -265. 

Express  as  a  decimal  6%4. 

Solution.—  6%4  =  64^4  _  ^4  =  x  _  <OIS6  =  .984. 

Express  as  a  decimal  1^64- 

Solution.—  1^4  =  %4  +  %4  =  M  +  %4  =  -125  +  -046 
=  .171. 

27 


28 


INDUSTRIAL   ARITHMETIC 


1.  Express  the  decimal  equivalent  to  three  places  of  each  of 
the  following: 


2.  What  error  is  made  in  considering  %Q  =  .186? 


3.  Obtain  a  piece  of  cardboard  or  stiff  paper.  Rule  it  as 
shown  below  making  the  left-hand  section  long  enough  to 
contain  the  64ths  from  i  to  32  and  the  right-hand  section 
the  64ths  from  33  to  64.  Fill  in  the  blanks  as  illustrated. 
Carry  the  decimals  to  six  places.  Keep  it  for  future  use. 


64 

ths 

11 

16 

ths 

8 
ths 

ths 

Dec. 

64 
ths 

I2 
ds 

16 

ths 

8 

ths 

ths 

Dec. 

I 

.015625 

33 

•515625 

2 

I 

.031250 

4 

2 

t 

.062500 

16 

8 

4 

2 

i 

.  250000 

LESSON  XIII 
REVIEW 

1.  In  the  formula  W  =  F.S.,  F  =  240  and  S  =  25.     Find  W. 

2.  If  H.P.  =  33^  and  V  =  16%  find  the  value  of  E  in  E  = 
33,000  H.P. 

V 

WV2 

3.  Given  that  T  =  -    — ,  find  the  value  of  T  when  W  = 

32.2 

16%  and  V  =  26. 

4.  The  barrel  reading  of  the  micrometer  when  used  to 
measure  the  diameter  of  a  piece  of  round  stock  is  4  large 
divisions  and  i  small  division.     What  is  the  diameter  of  the 
stock  expressed  as  a  common  fraction? 

5.  What  is  the  pitch  and  also  the  lead  of  each  of  the  following 
screws : 

25  threads  per  in.;  14;  18;  22;  38? 

6.  With  a  scale  of  %  in.  =  o  ft.  i  in.,  what  is  the  length  of 
the  drawing  to  represent  a  line  3  ft.  5^  in.;  8  ft.  6%  in.;  7.5 
in.;  %  in.? 

7.  The  fractional  dimensions  on  the  drawings  for  a  machinist 
are  usually  expressed  as  decimals.     Express  the  following  di- 
mensions suitable  for  a  machine  drawing: 

in.;  3^4^.;  5%  in.;  5%4  in. 


FIG.  20. 

8.  If  in  the  above  figure  AB  =  12  ft.  6  in.,  BC  =  3  ft.  9  in., 
CD  =  4  ft.  6  in.,  what  is  the  length  of  AD? 

29 


30  INDUSTRIAL   ARITHMETIC 

9.  If  AD  =  10  ft.;  AB  =  2  ft.  3  in.  and  CD  =  5  ft.  9  in.; 
BC  =  ? 

10.  If  BC  =  2AB  and  CD  =  3AB  and  AD  =  18  ft.,  find 
AB,  BC  and  CD. 

11.  If  BC  =  3.125  in.,  CD  =  4.265  in.  and  AD  =  12  in.; 
AB  =  ? 


LESSON  XIV 
AREA  OF  A  RECTANGLE 

Measuring  a  surface  consists  in  finding  the  number  of  squares 
the  surface  contains. 

There  are  many  standard  units  of  surface  in  common  use, 
such  as  the  square  inch,  square  foot,  square  yard,  etc. 

Problems 

1.  Find  the  number  of  square  feet  in  a  rectangle  of  base  10 
ft.,  altitude  4  ft. 

How  many  square  feet  in  the  row  along  the  ba^e?  How 
many  such  rows?  How  many  square  feet  in  the  rectangle? 


70 

FIG.  21. 

2.  Find  the  number  of  square  feet  in  each  of  the  following 
rectangles;  also  explain  as  in  problem  i: 

Altitude  5  ft.,  base  12  ft.;  altitude  18  ft.,  base  10  ft.;  alti- 
tude 4^  ft.,  base  18  ft.;  altitude  3^  ft.,  base  9  ft.;  altitude 
4  ft.,  base  8  ft.  6  in.;  altitude  4  ft.,  base  8  ft.  6  in.;  base  16  ft., 
altitude  5  ft.  3  in.;  altitude  5  ft.  6  in.,  base  24  ft.;  base  10  ft. 
10  in.,  altitude  12  ft.  6  in.;  base  b  ft.,  altitude  c  ft.;  base  x  ft., 
altitude  y  ft.;  base  m  ft.,  altitude  n  ft.;  base  k  ft.,  altitude  r  ft. 

3.  State  the  principle  you  have  used  in  solving  each  of  the 
above.     Express  the  principle  as  a  formula. 

31 


32  INDUSTRIAL   ARITHMETIC 

The  number  of  squares  in  a  surface  is  called  the  area  of  the 
surface  in  terms  of  the  square.  The  area  of  problem  i  is  40 
sq.  ft. 

Exercises 

1.  Find  the  number  of  square  feet  in  the  floor  of  a  room  15 
ft.  X  14  ft. 

2.  How  many  square  feet  in  the  surface  of  a  floor  14  ft.  7  in. 
long  and  12  ft.  5  in.  wide? 

3.  Find  the  number  of  square  feet  in  the  ceiling,  floor  and 
walls  of  a  room  24  ft.  X  15  ft.  X  10  ft. 

4.  Find  the  number  of  square  feet  in  the  surface  of  a  corridor 
200  ft.  long,  15  ft.  high  and  12  ft.  wide. 

5.  Find  the  number  of  square  feet  in  the  doors  of  your  school- 
room; in  the  windows. 

6.  Measure  the  chalk  box  and  find  the  number  of  square 
inches  its  surface  contains. 

7.  If  the  height  of  your  schoolroom  is  15  ft.,  find  the  number 
of  square  feet  in  its  surface. 

8.  Find  the  number  of  square  feet  of  plastering  in  the 
schoolroom. 

9.  A  box  a  ft.  long,  b  ft.  wide  and  c  ft.  high  contains  how 
many  square  feet  in  its  surface? 

10.  Find  the  area  of  the  floor  plan  in  lesson  9. 

11.  Assuming  200  sq.  ft.  of  surface  can  be  covered  with  two 
coats  with  i  gal.  of  paint,  how  many  gallons  will  be  required 
to  paint  the  walls  of  your  schoolroom? 

12.  What  decimal  part  of  an  acre  in  a  building  lot  86  ft.  X 
155  ft.     lA  =  43,5°°  sq-  ft- 


LESSON  XV 
REVIEW 

1.  Find  the  area  of  the  square  one  of  whose  sides  is: 

5  ft.;  6  ft.;  7  ft.;  ioft.;  12  ft;  15  ft.;  20  ft;  25ft;  24ft. 
6  in.;  23  ft.  2  in.;  28  ft.  6  in.;  15  ft.  4  in. 

2.  If  a  is  the  side  of  a  square  and  S  is  its  area  what  formula 
will  represent  its  area? 

3.  Draw  on  the  blackboard  a  square  foot,  a  square  yard. 
Then  fill  in  the  following  table  and  learn  it : 

-  sq.  in.  in  i  sq.  ft. 
—  sq.  ft.  in  i  sq.  yd. 

4.  How  many  square  inches  in  8  sq.  ft.?    9  sq.  ft.?     15  sq. 
ft,  98  sq.  in.?     18  sq.  ft,  75  sq.  in.?     3  sq.  yd.,  4  sq.  ft.,  80 
sq.  in.?     50  sq.  yd.,  2  sq.  ft,  85  sq.  in.? 

5.  If  you  are  given  the  product  of  two  numbers  and  one  of 
the  numbers,  how  will  you  find  the  other  number? 

Is  the  area  of  a  rectangle  the  product  of  two  numbers? 

When  you  know  the  area  of  a  rectangle  and  the  length  of  its 
base,  how  will  you  find  its  altitude? 

Let  S  =  area  of  a  rectangle,  h  its  altitude  and  b  its  base; 
then  state  as  a  formula  your  answer  to  the  last  question.  Also 
state  the  formula  for  its  base  in  terms  of  S  and  H. 

6.  Find  the  base  or  altitude  of  each  of  the  following  rec- 
tangles : 

S  =  500  sq.  in.;  b  =  10  in.;  S  =  750  sq.  yd.;  h  =  15  yd. 
S  =  1200  sq.  ft;  h  =  20  ft;  S  =  750  sq.  yd.;  h  =  15  ft. 
S  =  1728  sq.  ft;  b  =  12  ft;  S  =  750  sq.  yd.;  b  =  24  in. 

7.  How  wide  must  a  floor  35  ft.  long  be  to  contain  700  sq.  ft.? 

8.  I  wish  to  cut  from  a  board  8  in.  wide  a  piece  containing 
370  sq.  in.    How  long  a  piece  must  I  cut? 

3  33 


34  INDUSTRIAL   ARITHMETIC 

9.  If  20  workmen  are  placed  at  a  bench  3  ft.  wide,  all  on  the 
same  side  of  it,  how  long  must  the  bench  be  if  each  workman  is 
allowed  12  sq.  ft.? 

10.  Find  the  length  of  the  perimeter  of  a  rectangle  if  its  area 
is  ab  sq.  ft.,  and  its  altitude  is  a  ft.? 


LESSON  XVI 
SQUARE  ROOT 

The  square  root  of  a  given  number  is  the  number,  which  when 
squared  or  multiplied  by  itself,  will  produce  the  given  number; 
e.g.,  4  is  the  square  root  of  16,  because  42  =  16. 

Exercise. — What  is  the  square  root  of  each  of  the  following: 

25;  100;  144;  36;  64;  16;  81;  169;  2%6;  4%i;  10%44; 
16%25;  H;  A2;  B2;  N2;  P2? 

Numbers  whose  square  roots  can  be  expressed  by  means  of 
the  digits  only  are  called  perfect  squares. 

Exercise. — Name  all  the  perfect  squares  between  o  and  300. 

The  square  root  of  a  number  not  a  perfect  square  is  approxi- 
mately expressed  by  means  of  decimal  fractions.  Thus 
•\/5  =  2.236  approximately. 

Between  what  two  whole  numbers  does  the  square  root  of 
each  of  the  following  numbers  lie: 

26;  38;  45;  136;  175;  52;  70;  84? 

Exercises 

1.  Find  the  square  root  of: 

i369;  3!36;  65,536;  98,596;  277,729;  4096;  11,664;  6561; 
9216;  9,339,136;  7,387,524;  1,827,904;  .0841;  .061009;  4907- 
.0025. 

2.  Find  the  side  of  a  square  whose  area  is  225  sq.  ft. 

3.  I  desire  to  make  a  square  platform  whose  area  is  600 
sq.  ft.     Find  the  length  of  the  side  of  the  platform  so  that  the 
error  shall  be  less  than  ^  in. 

4.  It  is  required  to  construct  a  square  opening  in  the  walls 
of  a  room  for  a  ventilator.    The  opening  must  contain  600 

35 


36  INDUSTRIAL   ARITHMETIC 

sq.  in.     Find  the  side  of  the  opening  so  that  the  error  shall  be 
less  than  ^  in. 

5.  A  glazer  is  required  to  cut  a  pane  of  glass  in  the  form  of  a 
square  that  shall  contain  625  sq.  in.  What  size  must  he  cut 
the  glass? 


1.  Which  determines  the  shape  of  a  figure,  2  sq.  ft.  or  2  ft. 
square?     5  sq.  in.,  or  5  in.  square? 

2.  What  is  the  difference  in  square  feet  between  10  square 
feet  and  a  lo-ft.  square? 

What  is  the  difference  in  square  inches  between  a  6-in.  square 
and  6  sq.  in.? 

3.  How  long  a  board  will  it  take  to  fill  up  a  rectangular  hole 
of  160  sq.  in.  and  10  in.  wide? 

4.  How  many  boards  10  in.  wide  and  12  ft.  long  will  be 
required  to  build  a  solid  board  fence  5  ft.  high  and  48  ft.  long  if 
the  boards  are  nailed  lengthwise? 

6.  It  is  required  to  build  a  walk  96  ft.  long  and  2  ft.  8  in. 
wide  of  boards  16  ft.  long  and  10  in.  wide.  How  many  boards 
will  be  required  if  laid  lengthwise  and  how  wide  a  space  must 
be  left  between  the  boards? 

6.  In  a  room  1  2  f  t.  X  15  ft.  is  a  rug  9  ft.  X  1  2  ft.     How  many 
square  feet  of  floor  between  the  rug  and  the  walls  of  the 
room? 

7.  How  many  square  inches  in  a  picture  frame  15  in.  X 
9  in.  if  the  frame  is  2  in.  wide?     2^5  in.  wide?     2^  in.  wide? 

8-  15^  sq.  ft.  =  sq.  in.?  18%  sq.  ft.  =  sq.  in.? 
9%  sq.  yd.  =  sq.  ft.?  17}^  ft.  =  in.? 

9.  Extract  the  square  root  of  2,  of  3,  carry  the  result  to  three 
decimal  places. 

10. 


37 


LESSON  XVIII 
FRACTIONAL  REVIEW 


X  4M  X  X  =  ?         51A  X  3l/3  X 
X  3M  X  %  =  ?          SH  X  6%  X  5 

2.  H2  X  14%  X  %  =  ?       17^  X  5^  X 

i44  X  sK  X  ?H  =  ?       1728  X  4H  X  7^  =  ? 

3.  (^  +  %  +  %)  -  (M  +  K  +  Ms)  =  ? 

4.  (%)2  X  (%)3  X  (%)4  =  ? 

„   , 

—     * 


i      q  x 

i    x8 

+  ^  +  %)  x  (H  +  ^  +  Ho) 


8.  Ho  X  i%5  X  K4  X  i%7  =  ? 

9.  (4%5  +  8%l  +  4Ki) 

M  +  Hi      7^\ 


x  iH  +  H)  of  («K  +  ^s  x  2) 


of  2y3  of  2K)  -  (iK  of  i 

12.  What  number  multiplied  by  %i.  of  ^4  of  29}^  will  give 
102%  for  the  product? 

13.  If  the  rent  of  5^2  acres  of  land  is  $21%,  what  will  be 
the  rent  of  19^5  acres  of  the  land  at  the  same  rate? 

14.  How  many  taper  shanks  each  2^  in.  long  can  be  cut 
from  33^  ft.  of  stock  allowing  %  in.  to  each  piece  for  waste 
in  cutting? 

38 


LESSON  XIX 
AREA  OF  PARALLELOGRAMS 
F         A f         B 


FIG.  22. 

1.  Draw  a  parallelogram  ABCD  on  a  piece  of  cardboard  or 
stiff  paper.     Draw  CE  perpendicular  to  AB.     Cut  off  the 
triangle  CEB  and  place  in  the  position  of  FAD.     What  is  the 
name  of  the  parallelogram  CEFD  ?     Has  it  the  same  number  of 
squares  as  the  parallelogram  ABCD?    Has  it  the  same  base 
and  altitude?     How  do  you  find  the  number  of  squares  in 
CEFD  ?    How  then  in  AB  CD  ? 

2.  Find  the  number  of  square  units  in  each  of  the  following 
parallelograms: 

Base  10  ft.,  altitude  8  ft.;  altitude  17  ft.,  base  20  ft.;  base 
10  ft.,  altitude  8%  in.;  altitude  64  ft.,  base  25  ft.;  base  16^ 
in.,  altitude  8%  in.;  altitude  75  ft.,  base  33^  ft.;  base  ioft., 
altitude  5  ft.  9  in.;  altitude  13  ft.  8  in.,  base  12  ft.  6  in.;  base 
b  ft.,  altitude  a  ft.;  altitude  h  ft.,  base  m  ft. 

3.  The  expression  ab  sq.  ft.  stands  for  the  number  of  square 
feet  of  what  two  figures? 

4.  Find  the  missing  parts  of  the  following  parallelograms: 
40  sq.  ft.,  altitude  10  ft.;  75  sq.  ft.,  base  8  ft.;  125  sq.  ft., 

altitude  15  ft.;  altitude  18  ft.,  180  sq.  ft. 

39 


;      LESSON  XX 
AREA  OF  TRIANGLES 

1.  A  triangle  is  a  polygon  of  three  sides.  Any  one  of  the 
sides  is  a  base  of  the  triangle.  The  vertex  of  any  angle  is  a 
vertex  of  the  triangle.  How  many  bases  has  a  triangle? 
How  many  vertices? 

The  perpendicular  from  any  vertex  to  the  opposite  side  is  an 
altitude  of  the  triangle.  How  many  altitudes  has  a  triangle? 
Draw  a  triangle  and  its  altitudes. 


FIG.  24. 


2.  The  line  BD  is  called  a  diagonal  of  the  parallelogram. 
Name  another  diagonal  of  the  figure. 

3.  Draw  on  and  cut  out  from  a  piece  of  cardboard  or  stiff 
paper  a  parallelogram  ABCD.     Then  cut  along  the  diagonal 
BD.     Are  the  two  triangles  formed  equal?     A  triangle  with 
the  same  base  and  altitude  as  the  parallelogram  is  what  part  of 
the  parallelogram?     How  then  will  you  find  the  number  of 
square  units  in  a  triangle  when  its  base  and  altitude  are  given? 

Express  as  a  formula  your  answer  to  this  question. 

4.  Find  the  number  of  square  units  in  the  following  triangles: 
base  10  ft.,  altitude  8  ft.;  altitude  12  ft.,  base  20  ft.;  base  15 
ft.,  altitude  30  ft.;  altitude  18  ft.,  base  8  ft.;  base  12  ft.  6  in., 

40 


AREA   OF   TRIANGLES  41 

altitude  10  ft.;  altitude  18  ft.  8  in.,  base  10  ft.  6  in.;  base  b  ft., 
altitude  r  ft.,  base  x  ft.,  altitude  r  ft. 

5.  The  gable  of  a  house  is  30  ft.  wide  and  15  ft.  high.     How 
many  square  feet  in  the  two  gables? 

6.  How  many  square  feet  in  the  sides,  ends  and  two  gables 
of  a  house  40  ft.  long,  28  ft.  wide,  20  ft.  high  if  the  gables  are 
12  ft.  high? 


LESSON  XXI 
REVIEW 

1.  Find  the  number  of  square  feet  in  the  surface  of  15  steps 
if  the  height  of  the  risers  is  8  in.,  the  width  of  the  tread  is  10  in., 
and  the  width  of  the  steps  is  5  ft. 


FIG.  25. 

2.  Find  the  number  of  square  feet  in  this  figure,  scale 
=  i  ft.  o  in.  and  AB  =  5  ft. 


n. 


3.  Find  the  number  of  square  yards  of  canvas  required  to 
make  a  tent  in  the  shape  of  the  above  figure  if  AB  =  16  ft., 

42 


REVIEW  43 

CE  =  6  ft.  and  CD  =  20  ft.,  CB  =  10  ft.,  allowing  for  waste 
and  seams  one-fourth. 

4.  How  much  siding  will  be  necessary  to  side  a  house  32  ft. 
X  40  ft.  X  1 8  ft.  high  if  there  are  20  windows  3  ft.  X  7  ft. 
and  4  door  openings  3  ft.  6  in.  X  7  ft.  long  allowing  one-fourth 
of  the  net  area  for  waste? 

5.  A  table  top  is  6  ft.  4^  in.  long,  3  ft.  2  in.  wide  and  i%  in. 
thick.     How  much  surface  has  it,  not  including  underside? 

6.  How  many  shingles  will  be  needed  to  cover  a  pitch  roof 
40  ft.  long,  with  rafters  14  ft.  in  length,  allowing  1000  shingles 
for  each  100  sq.  ft.  of  surface? 

7.  What  is  the  cost  of  labor  upon  a  $2000  house,  if  the 
material  cost  $896,  excavation  and  cellar  $84  and  painting  $55? 

8.  If  r  =  1 6,  h  =  33  find  the  value  of  ?rr2h. 

9.  If  the  scale  is  i  in.  =  3  ft.  6  in.,  what  length  on  the  draw- 
ing to  represent  12  ft.  5  in.?     9  ft.  7  in.?     8  ft.  4  in.? 

10.  If  5.2  is  taken  for  the  square  root  of  27  show  that  the 
error  is  less  than  .004. 


LESSON  XXII 
REVIEW 

1.  Find  the  side  of  a  square  that  has  the  same  area  as  a 
parallelogram  15  ft.  X  12  ft.  6  in.  Give  result  correct  to  three 
decimal  places. 


1.2S- 


FIG.  27. 


FIG.  28. 


o  n 
6% 


^ 

OD 


i   i  /4 

FIG.  29. 

2.  How  many  rafters  are  required  for  a  pitch  roof  40  ft. 
long  if  the  rafters  are  set  2  ft.  to  centers? 

44 


REVIEW  45 

3.  Give  the  decimal  part  of  an  inch  of  each  of  the  following 
micrometer  readings: 

Four  large  and  2  small  of  the  barrel  and  8  of  the  thimble;  6 
large,  i  small  of  the  barrel  and  18  of  the  thimble;  2  large  barrel 
and  3  of  the  thimble;  2  small  barrel  and  2  thimble. 

4.  What  is  the  pitch  and  the  lead  of  a  screw  with  15  threads 
to  the  inch?     What  is  the  micrometer  reading  of  each? 

5.  The  base  of  a  triangle  is  28  ft.  8  in.     What  is  its  altitude 
if  it  has  the  same  area  as  a  rectangle  10  ft.  3  in.  X  24  ft.  6  in.? 

6.  Find  the  area  of  the  shaded  part  of  Fig.  27;  also  of  the 
I  section  (Fig.  28). 

7.  Find  the  area  of  the  part  of  the  rectangle  not  included 
within  the  small  rectangles  of  Fig.  29. 

8.  Compute  the  area  of  the  following  figure;  c  is  perpen- 
dicular to  a  and  b,  and  a  and  b  are  parallel: 


FIG.  30. 

a  =    10  ft.  3  in.;  b  =  15  ft.  6  in.;  c  =  4  ft.  5  in. 
9.  Find  the  side  of  the  square  that  has  the  same  area  as  the  I 
section  of  problem  6;  also  the  figure  of  problem  8. 


LESSON  XXIII 
CIRCLES 

A  circle  is  a  plane  figure  such  that  all  the  straight  lines  from  a 
point  to  a  closed  line  are  equal. 

The  point  is  called  the  center  of  the  circle. 

The  closed  line  is  called  the  circumference  of  the  circle. 

Any  one  of  the  straight  lines  from  the 
center  to  the  circumference  is  called  a 
radius  (plural  radii). 

A  straight  line  between  any  two  points 
of  the  circumference  is  called  a  chord. 

A  chord  through  the  center  is  called  the 
diameter  of  the  circle. 

Any  part  of  the  circumference  between 
two  of  its  points  is  called  an  arc. 

Name  the  center  of  the  above  circle.  A  radius,  a  chord,  an 
arc,  a  diameter. 

Is  a  diameter  a  chord?  Are  all  chords  diameters?  Are  any 
chords  diameters? 

A  radius  is  what  part  of  the  diameter  of  the  circle? 

If  the  radius  is  5.5  in.,  what  is  the  diameter?  If  the  diame- 
ter of  a  circle  is  18,  what  is  the  radius? 

Exercises 

1.  It  is  proved  in  geometry  that  the  circumference  of  a  circle 
is  equal  to  twice  it?  radius  multiplied  by  IT  or  C  =  27rR. 

Note  on  TT.  The  number  of  times  C  contains  D  (diameter)  is 
designated  by  the  Greek  letter  TT.  This  value  cannot  be  exactly 
expressed  by  means  of  the  digits,  but  correct  to  15  decimal 
places: 

TT  =  3.141592653589793 
46 


CIRCLES  47 

In  actual  work  the  value  we  shall  give  TT  depends  upon  the 
degree  of  accuracy  needed.  In  some  work  TT  =  3  will  give  a 
sufficiently  accurate  result,  in  other  work  TT  may  equal  3^7,  etc. 
In  most  ordinary  calculations  TT  =  3.1416  is  used. 

2.  The  number  of  square  units  in  a  circle  is  equal  to  the 
square  of  the  radius  mulliplid  by  IT  or  S  =  TrR2. 

3.  Find  the  circumference  of  each  of  the  following  circles: 
R  =  15;  16;  12;  10;  9^;  15^;  12^; 

D  =  24540;  16;  19;  31;  7;  8>£. 

4.  Find  the  number  of  square  units  in  each  of  the  circles  in 
problem  3. 

6.  Find  the  number  of  square  units  in  the  ring  between  two 
concentric  circles  of  radius  10  and  8  respectively. 

6.  A  circular  walk  4  ft.  wide  around  a  flower  bed  20  ft.  in 
diameter.     Find  the  number  of  square  feet  in  the  walk. 

7.  Inscribed  within  a  square  is  a  circle  whose  diameter  is 
10  ft.     Find  the  number  of  square  feet  between  the  circle  and 
the  sides  of  the  square. 

8.  If  the  inner  side  of  a  circular  running  track  is  %  mile  and 
the  track  is  20  ft.  wide  what  is  the  length  of  the  outer  side  of 
the  track? 

9.  If  the  diameter  of  a  circle  is  increased  by  i  ft.,  how  much  is 
the  circumference  increased? 

10.  The  diameter  of  the  earth  is  about  8000  miles  at  the 
equator.     Suppose  an  iron  band  lying  everywhere  upon  the 
equator  were  stretched  so  that  it  would  be  eyerywhere  %  ft. 
from  the  surface  of  the  earth,  how  many  feet  would  its  length  be 
increased? 

11.  If  the  radius  of  a  circle  were  multiplied  by  2,  by  3,  by  4, 
by  5,  by  6,  by  7,  by  8,  by  9,  by  10,  by  what  number  would  its 
circumference  be  multiplied?     Its  area? 

12.  What  is  the  number  of  square  feet  in  the  surface  of  the 
track  in  problem  9? 


LESSON  XXIV 
SPEED 

Speed  is  rate  of  change  of  position.  Speed  is  measured  by  a 
number  of  units  of  length  per  unit  of  time.  For  example, 
a  body  moving  through  1 20  ft.  each  minute  has  a  speed  of  1 20 
ft.  per  min.  or  2  ft.  per  sec.,  or  7200  ft.  per  hr. 

Exercises 

1.  A  point  is  moving  at  a  speed  of  i  mile  per  min.     What  is 
its  speed  per  hour?    Per  second? 

2.  What  is  the  speed  per  minute  of  a  train  running  60  miles 
per  hr.?     58  miles  per  hr.?     63  miles  per  hr.? 

3.  If  the  circumference  of  a  wheel  is  31.4  ft.,  what  is  the 
speed  of  a  point  on  its  rim,  if  the  wheel  is  making  100  .R.P.M. 
(revolutions  per  minute)  ? 

4.  What  is  the  speed  of  a  point  on  the  rim  of  a  cast-iron 
flywheel  i  ft.  o  in.  in  diameter,  when  the  wheel  is  making  1680 
R.P.M.? 

5.  A  lathe  spindle  is  running  1500  R.P.M.     What  is  the 
speed  of  a  point  on  the  surface  of  a  6-in.  cylinder  placed  in  the 
chuck? 

6.  Ordinarily  the  maximum  safe  rim  speed  of  cast-iron  fly- 
wheel with  solid  rim  is  about  85  ft.  per  sec.     Determine  if 
any  of  the  following  are  exceeding  the  safe  speed: 

i  ft.  diam.  making  2000  R.P.M.;  15  ft.  diam.  making  90 
R.P.M.;  6  ft.  diam.  making  280  R.P.M.;  7^  ft.  diam.  making 
220  R.P.M.;  9  ft.  diam.  making  185  R.P.M.;  8^  ft.  diam. 
making  200  R.P.M. 

48 


SPEED  49 

7.  Speeds  for  grindstones: 
Machinist's  tools,  800-1000  ft.  per  min. 
Carpenter's  tools,  500-600  ft.  per  min. 

The  maximum  safe  speed  for  a  grindstone  is  ordinarily  about 
3400  ft.  per  min. 

For  grinding  machinist's  tools  how  many  R.P.M.  should  a 
grindstone  3  ft.  in  diameter  make?  For  carpenter's  tools? 
What  is  the  maximum  safe  number  R.P.M. ? 

8.  If  S  =  surface  speed  of  a  point  on  a,  the  circumference  of  a 
revolving  wheel  or  cylinder,  D  =  the  diameter  of  the  wheel  or 
cylinder  and  R.P.M.  =  number  of  revolutions  per  minute  of 
the  wheel  or  cylinder,  express  the  formula  for  S  in  terms  of  D 
and  R.P.M. 

9.  Use  the  formula  derived  in  problem  8  and  find  S  when 

D  =  10  and  R.P.M.  =  20;  D  =  5;  R.P.M.  =  100;  D  = 
8;  R.P.M.  =  1200;  D  =  20;  R.P.M.  =  950. 

10.  If  S  =  500,  D  =  5;  find  R.P.M.;  if  S  =  1000,  R.P.M. 
=  500,  find  D;  if  S  =  2500,  R.P.M.  =  1500,  find  C;  if  S  = 
3000,    R.P.M.  =  1000,   find    C;  if   S  =  4000,    C  =  30,    find 
R.P.M.;  if  S  =  5000,  C  =  40,  find  R.P.M. 


LESSON  XXV 
SPEEDS  OF  PULLEYS  AND  GEARS 

If  two  pulleys  are  connected  by  a  belt  and  one  of  the  pulleys 
set  in  motion,  the  belt  will  cause  the  other  pulley  to  move  also, 
that  is,  motion  is  transmitted  from  one  pulley  to  the  other  by 
means  of  the  belt.  The  pulley  that  transmits  motion  to  the 
belt  is  called  the  driving  pulley.  The  pulley  that  is  set  in 
motion,  or  to  which  motion  is  transmitted,  by  the  belt  is  called 
the  driven  pulley. 


FIG.  32. 

Motion  is  also  transmitted  by  means  of  gears.  In  this  case 
the  motion  is  transmitted  by  direct  contact,  the  teeth  of  the 
driving  gear  being  made  to  mesh  with  the  teeth  of  the  driven. 
Should  it,  however,  happen  that  the  two  gears  are  too  far  apart 
to  mesh,  an  intermediate  gear  is  used  to  transmit  motion  from 
the  driving  gear  to  the  driven  gear.  Three  or  more  gears 
meshing  together  form  a  train  of  gears,  one  or  more  of  which  is 
always  an  intermediate.  Gear  B  is  the  intermediate. 

Speed  of  pulleys  or  gears  means  the  number  of  revolutions  per 
minute  (R.P.M.)  the  pulleys  or  the  gears  are  making. 

So 


SPEEDS   OF  PULLEYS   AND   GEARS  51 

Exercises 

1.  If  the  driving  gear  has  24  teeth  and  the  driven  gear  8, 
when  the  driving  gear  has  made  one  revolution,  how  many 
will  the  driven  gear  have  made? 

2.  Driving  gear  36  teeth,  driven  gear  9  teeth;  driving  gear 
40  teeth,  driven  gear  5  teeth,  driving  gear  18  teeth,  driven 
gear  7  teeth;  driving  gear  21  teeth,  driven  gear  6  teeth. 

How  many  revolutions  will  the  driven  gear  in  each  of  the 
above  make  for  one  revolution  of  the  driving  gear? 

3.  If  the  driving  gear  contains  32  teeth  and  is  making  40 
R.P.M.  what  is  the  speed  of  the  driven  gear  with  9  teeth? 

4.  The  front  sprocket  of  a  bicycle  contains  24  teeth  and* the 
rear  sprocket  8  teeth,  how  many  revolutions  will  the  pedals 
make  in  going  i  mile?     The  wheels  of  the  bicycle  being  28  in. 
in  diameter? 

5.  The  driving  gear  has  30  teeth  and  the  driven  gear  10  teeth. 
If  they  are  connected  with  an  intermediate  of  40  teeth,  what 
number  of  revolutions  does  the  driven  gear  make  for  each  of 
the   driving  gear?     Explain  why  this  is  true.     What  effect 
upon  the  speed  of  the  driven  gear  has  the  intermediate? 

6.  The  diameter  of  the  driving  pulley  is  1 2  in.  and  its  speed 
300  R.P.M.     What  is  the  speed  of  the  driven  pulley  whose 
diameter  is  4  in.;  3  in.;  5  in.? 

7.  The  diameter  of  the  driving  pulley  is  10  in.  and  its  speed 
900  R.P.M.     Required  the  speed  of  the  driven  pulley  of 
diameter  4  in. 

8.  On  the  driving  shaft  is  a  24-in.  pulley  making  300  R.P.M. 
What  is  the  speed  of  the  driven  shaft  which  has  a  lo-in.  pulley 
belted  to  the  driver? 

9.  Find  the  surface  speed  of  each  pulley  in  problem  8. 

10.  If  the  surface  speed  of  6-in.  pulley  is  3000  ft.  per  min. ; 
how  many  R.P.M.  is  it  making? 

11.  The  maximum  safe  surface  speed  of  a  grindstone  is 


52  •  INDUSTRIAL   ARITHMETIC 

2400  ft.  per  min.     Find  the  maximum  safe  number  R.P.M.  a 
6-ft.  stone  may  make. 

12.  If   the  surface  speed  of  an  emery  wheel  making  600 
R.P.M.  is  4000  ft.,  what  is  its  diameter? 

13.  A  driven  pulley  5  in.  in  diameter  has  a  speed  of  2500 
R.P.M.     If  the  speed  of  the  driving  pulley  is  500  R.P.M.,  what 
is  its  diameter? 

14.  The  speed  of  the  driving  pulley  is  N  =  R.P.M.  and  its 
diameter  D.     Find  the  formula  for  the  speed  S  of  the  driven 
pulley  whose  diameter  is  d. 

16.  Use  formula  of  problem  14  and  find  S  if  N  =  120, 
D  =  12,  and  d  =  6;  if  N  =  1200,  D  =  20,  d  =  5,  find  S;  if  S 
=  500,  d  =  10,  D  =  25,  find  N;  if  S  =  2500,  d  =  8,  D  =  24, 
find  N;  if  N  =  3000,  D  =  18,  d  =  5,  find  S;  if  N=  1800,  D 
=  40,  d  =  8,  find  S;  if  S  =  900,  D  =  20,  d  =  2,  find  N. 

16.  If  D  =  number  of  teeth  in  driving  gear,  d  =  in  driven 
gear  N  =  R.P.M.  of  D  and  n  =  R.P.M.  of  d,  show  that 

D  _  n 
d  ~N 

17;  Use  formula  of  problem  16  to  find  D  when  d  =  10, 
n  =  30,  N  =  60;  to  find  N  if  D  =  40,  d  =  24,  n  =  12;  to 
find  d  if  D  =  80,  N  =  4,  n  =  8;  to  find  D  if  N  =  12,  n  =  6 
and  d  =  20;  to  find  N  if  D  =  80,  d  =  28  and  n  =  2. 

18. 


FIG.  33. 

In  the  above  train,  E  and  F  are  keyed  to  the  same  shaft. 
D  has  60  teeth,  E  40,  F  30  and  G  45.  Find  the  number  of 
revolutions  of  G  for  each  revolution  of  D. 


SPEEDS   OF   PULLEYS   AND   GEARS 


53 


Suppose  an  intermediate  of  60  teeth  is  placed  between  F  and 
G,  then  what  is  the  result? 

19.  Let  D  =  100,  E  =  90,  F  =  70  andG  =  55.  If  R.P.M. 
of  D  =  40  find  R.P.M.  of  G. 

20. 


FIG.  34. 

E  and  F  are  keyed  to  the  same  shaft,  likewise  C  and  D. 
A  =  40,  B  =  20,  C  =  35,  D  =  50,  E  =  45,  F  =  55  and  G  = 
60.  For  each  revolution  of  A,  find  number  of  revolutions  of  G. 

21.  Let  G  =  80,  F  =  70,  E  =  60,  C  =  50,  D  =  55,6  =  20 
and  A  =  40.  When  G  has  made  one  revolution,  how  many  has 
A  made? 


LESSON  XXVI 
CUTTING  SPEED  AND  FEED 

When  turning  pr  cutting  a  cylindrical  piece  in  a  lathe  the 
number  of  lineal  feet  of  the  surface  of  the  piece  cut  by  the  tool 
in  i  min.  is  called  cutting  speed  of  the  tool.  For  example, 
if  the  piece  is  making  20  R.P.M.  and  its  circumference  is  4  in., 
the  cutting  speed  is  evidently  4  in.  X  20  =  80  in.  =  6%  ft.  per 
min. 

The  distance  the  tool  advances  along  the  work  in  each  revolu- 
tion is  called  feed.  It  is  expressed  in  ordinary  work  as  a 
fractional  part  of  an  inch.  For  example,  if  the  tool  advances 
^fg  m-  along  the  work  in  one  revolution  the  feed  is  H$  in. 
The  width  of  the  chip  is  equal  to  the  feed. 

Exercises 

1.  Is  the  surface  speed  per  minute  of  a  revolving  cylinder  the 
same  as  the  cutting  speed  of  the  tool?     Why? 

2.  A  piece  of  round  stock  2  in.  in  diameter  is  making  40 
R.P.M.    What  is  the  cutting  speed? 

3.  If  the  feed  is  ]/{Q  in.  and  the  work  has  a  speed  of  42 
R.P.M.  how  long  will  it  require  to  turn  a  piece  3  in.  long? 

4.  A  piece  of  work  revolves  50  times  while  the  tool  advances 
i%6".      Find  the  feed. 

6.  If  the  feed  is  ^2  m-  and  the  cutting  speed  20  ft.  per  min., 
find  the  time  required  for  the  tool  to  travel  i  in.  along  a  piece 
4  in.  in  diameter. 

6.  When  cutting  soft  steel  the  speed  may  be  100  ft.  per  min. 
if  the  depth  of  the  cut  is  ^  in.  and  the  feed  ^2  m-  Find  tne 

54 


CUTTING   SPEED   AND   FEED  55 

R.P.M.  a  cylinder  of  soft  steel  3  in.  in  diameter  is  making  from 
the  above  data. 

7.  If  the  feed  per  revolution  of  a  drill  is  .0075  in.,  find  the 
time  required  to  drill  a  hole  .25  in.  in  diameter  through  a 
rectangular  piece  of  steel  iY±  in.  thick,  if  the  drill  is  making  735 
R.P.M. 

8.  How  many  strokes  of  the  shaper  will  be  required  to  rough 
cut  a  rectangular  piece  2%  in.  wide,  if  the  feed  is  ^2  in.? 

9.  A  round  piece  36  in.  long  and  2  in.  in  diameter  is  to  be 
turned  to  a  diameter  of  i%  in.     How  long  will  be  required  to 
do  the  work  if  the  feed  of  the  rough  cut  is  34o  in.  and  the 
finish  cut  j^o  in.  and  the  work  has  a  speed  of  120  R.P.M. 

10.  If  F  =  feed,  N  =  R.P.M.  and  D  =  distance  the  tool 
moves  find  F  in  terms  of  D  and  N. 

11.  Use  the  formula  of  problem  10. 

If  F  =  Ho  in.,  R.P.M.  =  50,  find  D;  if  D  =  2,  F  =  M2 
in.,  find  R.P.M.;  if  R.P.M.  =  40,  F  =  %  in.,  findD;  if  D  =  i, 
R.P.M.  =  32,  find  F;"if  R.P.M.  =  32,  D  =  2,  findF;  if  D  = 
3,  R.P.M.  =  32,  find  F. 


LESSON  XXVII 
AREA  OF  SURFACE  OF  CYLINDER 

In  geometry  it  is  proved  that  the  area  of  the  curved  surface  of 
a  right  cylinder  is  the  circumference  of  its  base  multiplied  by  its 
altitude. 

The  complete  area  includes  the  area  of  the  curved  surface 
and  the  area  of  the  two  ends. 

Exercises 

1.  Find  the  number  of  square  units  in  the  curved  surface  of 
each  of  the  following  cylinders: 

Circumference  of  base  28  in., 
altitude  10  in.;  circumference  of 
base  38  ft.,  altitude  12  ft.;  cir- 
cumference of  base  42.5  ft.,  alti- 
tude   22    ft.;   circumference   of 
base  15  ft.  8  in.,  altitude  9  ft. 
2.  Find  the  number  of  square 
in  the  curved  surface  of 
each  of  the  following  cylinders: 
Altitude  18  ft.,  radius  of  base  10  ft.;  altitude  10  ft.,  radius  of 
base  8  ft.;  altitude  8  ft.,  radius  of  base  12  ft.;  altitude  15  ft., 
radius  of  base  9  ft.  6  in. 

3.  Find  the  number  of  square  units  in  the  complete  surface 
of  each  of  the  cylinders  in  problem  2. 

4.  Find  the  cost  of  painting  a  cylindrical  column  20  ft. 
high  and  3  ft.  in  diameter  at  5f£.  per  sq.  ft. 

5.  How  many  square  feet  of  tin  will  be  required  to  line  the 
bottom  and  side  of  a  cylindrical  tank  10  ft.  high  and  8  ft.  in 
diameter?     What  will  be  the  cost  at  ioj£.  per  sq.  ft.? 

6.  Express  the  formula  for  area  of  the  curved  surface  of 
a  cylinder;  the  complete  area  of  a  cylinder;  if  H  =  its  alti- 
tude and  R  =  radius  of  its  base. 

56 


FIG.  35. — Figure  of  a  cylinder 
showing  its  surface  changed  to  unj£S 
a  rectangle. 


LESSON  XXVIII 
REVIEW 

1.  Find  the  side  of  a  square  that  has  the  same  area  as  the 
complete  area  of  a  cylinder  2  ft.  in  diameter  and  4  ft.  long. 
TT  =  3.1416. 

2.  A  shaft  to  which  are  attached  two  pulleys  is  making  300 
R.P.M.     Compare  the  surface  speeds  of  the  two  pulleys  if 
their  diameters  are  respectfully  18  in.  and  10  in. 

3.  The  outside  diameter  of  a  pipe  is  2  in.  and  the  inside 
diameter  i%  in.     Find  the  difference  between  the  outside  and 
inside  areas  of  a  piece  10  ft.  long. 

4.  A  grinding  wheel  6  in.  in  diameter  has  a  surface  speed  of 
500  ft.  per  min.     The  wheel  is  attached  to  a  shaft  which  has  a 
3-in.  pulley,  which  is  belted  to  lo-in.  pulley  on  another  shaft 
on  which  is  also  a  4-in.  pulley;  this  pulley  is  belted  to  a  i2-in. 
pulley  on  another  shaft  on  which  is  another  i2-in.  pulley  which 
is  belted  to  a  5-in.  pulley  attached  to  a  shaft  run  directly  by 
the  motor.     Find  R.P.M.  of  motor. 

5.  Three  pipes  respectively  3  in.,  4  in.  and  6  in.  in  diameter 
are  discharging  into  a  header.     What  is  the  diameter  of  the 
header,  if  the  rate  of  flow  in  all  four  pipes  is  the  same  and  all 
the  pipes  are  full? 

6.  How  many  R.P.M.  must  a  cylinder  10  in.  in  diameter 
make  in  order  that  the  tool  may  advance  2  in.  in  3  min.,  if  the 
feed  is  %4  in.? 

7.  If  the  driving  gear  has  75  teeth  and  a  speed  of  a  40  R.P.M. 
what  is  the  speed  of  the  driven  gear  with  25  teeth? 

8.  How  will  you  set  the  micrometer  for  .149  in.?     .403  in.? 
.738  in.? 

57 


58  INDUSTRIAL   ARITHMETIC 

9.  How  many  revolutions  must  a  thread  whose  pitch  is 
make  to  advance  .2  in.?     %  in.? 

10.  If  B  =  36,  b  =  49  and  H  =  27,  find  V. 

V  =  ?  (B  +  b  +  VBb). 
o 

11.  If  R  =  12,  r  =  9  and  H  =  15,  find  V. 

V  =—  (R2  +  r2  +  Rr. 


LESSON  XXIX 


VOLUME  OF  A  PRISM 

1.  A  prism  is  a  solid  whose  bases  are  parallel  and  whose 
faces  are  parallelograms. 

2.  A  prism  takes  its  name  from  the  shape  of  its  base,  e.g., 
if  the  base  is  a  triangle,  it  is  called  a  triangular  prism;  if  a 
quadrilateral,  a  quadrangular  prism,  etc. 

3.  The    perpendicular    between    the 
bases  is  called  the  altitude  of  the  prism. 

4.  The  intersections  of  the  faces  are 
called  the  edges  of  the  prism. 

5.  If  the  edges  are  perpendicular  to 
the  bases  the  prism  is  a  right  prism. 
We   will   consider   only   right   prisms. 
The  edge  of  a  right  prism  is  equal  to 
the  altitude  of  the  prism. 

6.  A  prism  whose  bases  and  faces 
are  squares  is  called  a  cube. 

7.  Measuring  a   solid  consists  in  finding  the  number  of 
standard  cubes  a  solid  contains. 

A  standard  cube  is  a  cube  each  of  whose  edges  is  a  standard 
unit  of  length;  e.g.,  a  cubic  inch,  cubic  foot,  etc.,  is  a  cube  each 
of  whose  edges  is  an  inch,  a  foot,  etc. 

8.  The  number  of  cubes  a  solid  contains  is  called  its  volume 
in  terms  of  the  cube;  e.g.,  if  a  solid  contains  15  cu.  ft.,  its 
volume  is  1 5  in  terms  of  a  cubic  foot. 

59 


FIG.  36. — Right  Prism. 


6o 


INDUSTRIAL   ARITHMETIC 


9.  Find  the  volume  of  a  prism  6  in.  long,  3  in.  wide  and  3  in. 
high.  How  many  cubes  in  a  section  along  the  base?  How 
many  in  one  layer?  How  many  in  three  layers? 


/  / 


xz 


AS 


\ 

A- 

A 

/T 
/  \ 

-    ,/r 

?r 
/  1 

/  1 

•     )• 

y 

__-J_ 

/ 

—  / 

/ 

FIG.  37. 


10.  Find  the  number  of  cubes  in  each  of  the  following 
prisms : 

8  in.  X  3  in.  X  4  in.;  15  in.  X  12  in.  X  6  in.;  9  ft.  X  9  ft.  X 
9  ft.;  7  ft.  X  7  ft.  X  10  ft;  18  ft.  X  20  ft.  X  8  ft;  17  ft.  X 
6  ft  X  4  ft;  10  ft.  X  8  ft  X  12  ft;  9  ft  X  8  ft  X  6  ft;  25 
ft.  X  15  ft.  X  20  ft;  10  ft.  6  in.  X  8  ft  6  in.  X  7  ft  3  in.; 
15  ft.  8  in.  X  12  ft.  9  in.  X  10  ft  6  in. 

11.  If  each  edge  of  a  cube  is  5  in.  how  many  cubic  inches  in 
it?     If  6  in.?     If  10  in.?     If  12  in.?     If  15  in.?     If  a  in.? 
If  b  in.?     If  c  in.?     If  n  in.? 

12.  How  many  cubic  inches  in  a  cubic  foot?     How  many 
cubic  feet  in  a  cubic  yard? 

Fill  in  the  following  table  and  learn  it. 

— cu.  in.  =  i  cu.  ft. 
cu.  ft.  =  i  cu.  yd. 

13.  Change  to  cubic  inches,  18  cu.  ft.;  25  cu.  ft.;  33^  cu. 
ft.;  12^  cu.  ft;  100  cu.  ft. 

14.  Change  to  cubic  yards,  5760  cu.  ft;  9000  cu.  ft.;  1765 
cu.  ft. 


VOLUME   OF   A   PRISM  6 1 

16.  Change  to  cubic  feet,  15,625  cu.  in.;  172,800  cu.  in.; 
2456  cu.  in. 

16.  The  dimensions  of  a  rectangular  prism  are  a,  b  and  c, 
derive  the  formula  for  its  volume. 

17.  If  a  =  4,       b  =  5,   c  =  3,      V  =  ? 

a  =  10,  b  =  9,  c  =  8,  V  =  ? 
a  =  8,  b  =  6,  c  =  12,  V  =  ? 

18.  If  V  =  144,  a  =  4,  b  =  9,       c=? 

V  =  250,  a  =  6,  b  =  10,  c  =  ? 
V  =  400,  b  =  5,  c  =  12,  a  =  ? 
V  =  920,  a  =  9,  c  =  10,  b  =  ? 


LESSON  XXX 
REVIEW 

1.  Find  the  number  of  cubic  yards  of  earth  taken  from  a 
cellar  27  ft.  long,  25  ft.  wide  and  5  ft.  deep. 

2.  A  ditch  i  mile  long,  10  ft.  wide  and  6  ft.  deep  had  how 
many  cubic  yards  of  earth  removed  from  it? 

3.  How  many  cubic  feet  of  masonry  in  a  wall  40  ft.  long,  15 
ft.  high  and  4  ft.  6  in.  wide? 

4.  A  wall  2  ft.  thick  and  8  ft.  high  is  built  around  a  lot  50 
ft.  X  150  ft.     Find  the  number  of  cubic  yards  of  masonry  in  it 
if  there  is  an  opening  10  ft.  wide  in  the  wall.     (Two  solutions.) 

5.  If  your  schoolroom  is  1 2  ft.  high,  how  many  cubic  feet  of 
air  are  in  the  room? 

6.  A  piece  of  lumber  16  ft.  and  2  in.  X  8  in.  contains  how 
many  cubic  feet  of  wood? 

7.  What  will  be  the  cost  of  a  house  36  ft.  X  25  ft.  and  20  ft. 
high  at  20fi.  per  cu.  ft.? 

8.  A  tank  10  ft.  square  and  12  ft.  deep  will  hold  how  many 
gallons  of  water?     i  cu.  ft.  contains  7^  gal. 

9.  How  many  cubic  feet  of  wood  in  a  pile  8  ft.  long,  4  ft. 
wide  and  4  ft.  high? 

10.  Find  the  number  of  cubic  feet  in  a  box  5  ft.  6  in.  long, 
3  ft.  8  in.  wide  and  2  ft.  9  in.  high. 

11.  Use  the  formula  for  the  volume  of  a  prism  and  solve 
for  a  in  terms  of  the  other  letters,  likewise  for  b,  for  c. 

12.  State   in   words   the   results   you   have   obtained    in 
problem  n. 


62 


LESSON  XXXI 
REVIEW 

1.  A  piece  of  steel  %  in.  square  and  7.25  in.  long  is  taken  to 
make  a  lathe  tool.     Find  its  weight  if  i  cu.  in.  of  steel  weighs 
.28  Ib. 

2.  If  the  piece  in  problem  i  weighs  4  Ib.,  what  was  its 
length? 

3.  The  base  of  a  rectangular  prism  is  8  ft.  X  5  ft.     What  is 
its  height,  if  it  has  the  same  volume  as  a  6-in.  cube? 

4.  A  cube  of  steel  6-in  edge  is  hammered  when  hot  into  a 
rectangular  prism  whose  base  is  5.6  in.  X  3.4  in.     Find  the 
length  of  the  prism. 

5.  A  rectangular  piece  of  steel  8  in.  X  5  in.  X  3.5  in.  has 
each   dimension  reduced   %  in.     How   much  is  its  weight 
reduced? 

6.  A  liquid  gallon   contains    231    cu.    in.     A   rectangular 
measure  4  in.  square  must  be  how  deep  to  hold  i  qt.  ?     i  gal.  ? 

7.  If  the  I  section  of  problem  6,  lesson  22,  is  8.25  ft.  long, 
what  is  its  volume? 

8.  A  rectangular  tank  15  ft.  long  and  12  ft.  wide  contains  500 
gal.  of  water.     How  deep  is  the  water  in  feet? 

9.  If  the  weight  of  the  water  in  the  tank  of  problem  8  weighs 
i  ton,  how  deep  is  the  water?     i  cu.  ft.  weighs  62.5  Ib. 

10.  Two  rectangular  pieces  of  metal  5  ft.  X  5  ft.  X  8  ft. 
and  6.5  ft.  X  3.4  ft.  X  10  ft.  are  melted  and  cast  into  a  single 
rectangular  piece  8  ft.  square.     How  long  is  the  new  piece? 

11.  A  bar  of  iron  2  in.  square  and  i  in.  long  is  drawn  out 
until  it  is  1^3  in.  X  i  in.     What  is  its  length? 

12.  If  a  piece  of  stock  2^  in.  square  and  8  in.  long  is  forged 
into  a  piece  2  in.  square,  how  long  is  the  new  piece? 

63 


64  INDUSTRIAL   ARITHMETIC 

13.  The  sides  and  bottom  of  an  open  steel  tank  4  ft.  square 
and  6  ft.  high  outside  dimensions  are  2  in.  thick,  find  the 
weight  of  the  tank. 

14.  A  piece  of  steel  14.5  in.  X  16  in.  and  2  in.  thick  has  a 
rectangular  hole  8  in.  X  4.25  in.  cut  in  it.     What  is  the  weight 
of  the  piece  of  steel? 

15.  If  the  scale  is  i"  =  3'  4"  how  long  must  the  drawing 
be  for  each  of  the  following  lengths: 

12  ft.  o  in;  15  ft.  7  in;  16  ft.  9  in.;  7  ft.  10  in.;  23  ft.  8  in.? 


LESSON  XXXII 
REVIEW  OF  PERCENTAGE 

The  expression  6%  means  .06  or  *Koo- 
The  expression  28.3%  means  .283  or  283/Looo- 
25%  =  2^00  =  M;  33^%  =  33Vloo  =  1A\  20%  =  ?/ioo 
=  ?5  50%  -  ?  -  ? 

Exercises 

1.  Express   as  a  decimal  fraction   without  their  denomi- 
natots  each  of  the  following: 

12^%;  32%;  35-3%;  96-4%;  76.5%;  87.7%. 

2.  25%  of  144  =  ?     12%%  of  840  =  ?  20%  of  255  =  ? 
33^%  of  175  =  ?     75%  of  164  =  ?     66%%  of  930  =  ? 

3.  28%  of  what  number  is  28? 
4-  73%  °f  what  number  is  146? 

6.  86.2%  of  what  number  is  2586? 

6.  28  is  what  per  cent,  of  560?    35  is  what  per  cent,  of  70? 

7.  144  is  what  per  cent,  of  770?     29  is  what  per  cent,  of  125? 

8.  How  many  board  feet  of  flooring  must  be  purchased  for  a 
floor  25  ft.  X  1 8  ft.  if  an  allowance  of  20%  is  added  for  waste? 

9.  In  finishing  a  piece  of  steel  its  weight  was  reduced  1.5%. 
What  was  the  weight  of  the  finished  piece  if  the  rough  piece 
weighed  78.31  lb.? 

10.  How  many  pounds  of  lead  are  there  in  168  lb.  of  soft 
solder,  if  33^%  the  solder  is  lead? 

11.  A  certain  grade  of  steel  contains  3.4%  nickel.     How 
many  pounds  of  nickel  in  i  ton  of  this  grade  of  steel? 

12.  The  foot  shrink  rule  used  by  the  pattern  maker  is  12^ 
in.  in  length.     The  carpenter's  foot  rule  is  what  per  cent,  of 
the  length  of  the  pattern  maker's  foot  rule? 

13.  About  .37%  of  soft  steel  is  silicon.     How  much  silicon 
is  there  in  1500  lb.  of  soft  steel? 

5  65 


LESSON  XXXIII 
BOARD  MEASURE 

ft 

1.  A  board  foot  is  a  piece  of  lumber  having  an  area  of  i 
sq.  ft.  on  its  flat  surface  and  thickness  of  i  in.  or  less.     A  board 
foot  i  in.  thick  contains  how  many  cubic  inches? 

2.  In  estimating  the  number  of  board  feet  in  a  piece  of  wood 
it  is  customary  to  estimate  thickness  less  than  i  in.  as  an  entire 
inch;  e.g.,  a  board  12  ft.  long,  8  in.  wide  and  %  in.  thick  is 
estimated  the  same  as  if  it  were  1 2  ft.  long,  8  in.  wide  and  i  in. 
thick. 

3.  Find  the  number  of  board  feet  in  a  piece  of  lumber  i^  in. 
thick,  8  in.  wide  and  16  ft.  long. 

4.  Find  the  number  of  board  feet  in  a  piece  of  lumber  6  in. 
thick,  8  in.  wide  and  16  ft.  long. 

5.  Find  the  number  of  board  feet  of  lumber  in  each  of  the 
following: 

i  in.  X  4  in.  X  12  ft.;  i^  in.  X  8  in.  X  16  ft.;  5  in.  X  4  in. 
X  18  ft;  2  in.  X  6  in.  X  16  ft.;  %  in.  X  6  in.  X  20  ft.;  2  in. 
X  Qin.  X  i6ft.;7in.  X  i8in.  X  20  ft.;  7  in.  X  8  in.  X  12  ft.; 
i^  in.  X  6  in.  X  12  ft.;  %  in.  X  8  in.  X  16  ft.;  i%  in.  X 
10  in.  X  18  ft.;  4  in.  X  6  in.  X  14  ft. 

6.  How  many  board  feet  of  flooring  i  in.  thick  are  required 
for  a  floor  20  ft.  6  in.  X  15  ft.  9  in.  if  25%  is  allowed  for 
waste? 

7.  How  many  board  feet  in  10  pieces  of  lumber  2  in.  X  8  in. 
and  1 6  ft.  long. 


66 


LESSON  XXXIV 
REVIEW 

1.  The  outside  dimensions  of  a  box  are  4  ft.  X  3  ft.  X  2  ft. 
If  the  boards  are  i  in.  thick,  how  many  board  feet  are  in  the 
box  and  its  lid? 

2.  The  inside  dimensions  of  a  coal  bin  are  10  ft.  X  6  ft.  X  5 
ft.     If  the  sides  are  2  in.  thick,  how  many  board  feet  of  lumber 
are  in  its  sides,  if  one  of  the  sides  is  cellar  wall? 

3.  The  floor  of  a  veranda  is  laid  of  boards  i^  in.  thick.     If 
the  floor  is  8  ft.  X  23  ft.  6  in.,  how  many  board  feet  of  flooring 
does  it  require,  allowing  10%  for  waste.     Find  the  cost  at  $60 
per  M. 

4.  How  many  board  feet  of  lumber  will  be  required  to  floor  a 
platform  16  ft.  4  in.  long  X  n  ft.  6  in.  wide,  the  lumber  to  be 
i  in.  thick,  if  no  allowance  is  made  for  waste?     If  10%  is 
allowed  for  waste? 

5.  Find  the  area  (complete)  of  a  cylinder  10  ft.  high  and  10 
ft.  in  diameter. 

6.  How  many  square  feet  of  tin  will  be  required  to  make  a 
pipe  4  ft.  long  and  7  in.  in  diameter  if  i  in.  is  allowed  for  the 
seam?     Find  the  cost  at  lojif.  per  sq.  ft.    TT  =  3%. 

7.  If  a  wheel  2  in.  in  diameter  is  making  2500  R.P.M., 
through  how  many  feet  will  a  point  on  its  rim  pass  in  i  min.? 
In  i  hr.? 

8.  A  chimney  30  ft.  high  is  18  in.  square  and  has  a  flue  12  in. 
square.    How  many  cubic  feet  in  the  masonry  of  the  chimney? 

9.  If  the  feed  is  %  in.  and  the  work  has  a  speed  of  40  R.P.M. 
how  long  will  it  require  to  turn  a  piece  4^4  in.  long? 

10.  If  the  feed  of  the  shaper  is  ^2  m->  now  many  strokes 
must  it  make  to  rough  cut  a  piece  3^6  m  wide? 

67 


LESSON  XXXV 
WOODWORKING  PROBLEMS 

Rough  undressed  lumber  for  shopwork  is  sawed  into  the 
following  standard  thickness: 

i  in.;  \}^  in.;  i^  in.;  2  in.;  2}^  in.;  5  in.;  3^  in.;  and 
4  in. 

In  dressing  lumber  about  %  in.  of  its  thickness  is  removed, 
that  is  a  i-in.  board  when  dressed  will  be  %  in.  thick. 

If  a  dressed  board  i%  in.  in  thickness  is  wanted  a  2-in. 
thickness  must  be  dressed  down  Y±  in. 

In  ordering  lumber  from  the  mill  the  number  of  pieces 
wanted,  the  thickness,  the  width  and  length  of  each  piece 
must  be  given. 

A  piece  of  lumber  2  in.  X  2^  in.  X  40  in.  means  the  thick- 
ness is  2  in.,  the  width  2}^  in.  and  the  length  40  in. 

A  mill  bill  is  an  order  on  the  mill  for  lumber  wanted.  It 
must  contain  the  number  of  pieces,  kind  of  wood  and  dimen- 
sions of  each  piece.  The  dimensions  are  usually  given  for 
rough  lumber. 

In  estimating  the  amount  of  lumber  required  for  a  given  job, 
add  about  25%  of  the  actual  number  of  board  feet  in  the 
finished  article.  This  allowance  is  for  waste.  Waste  in 
handling  lumber  results  from  any  or  all  of  the  following  causes : 

1.  Saw  cuts  about  %  in. 

2.  Checks  or  cracks  in  lumber  especially  at  ends. 

3.  Knots. 

4.  Small  pieces  resulting  from  sawing. 

68 


WOODWORKING   PROBLEMS 


69 


5.  General  defects  in  lumber. 

6.  Dressing  of  lumber. 


Exercises 

1.  Find  the  total  number  of  board  feet  and  the  cost  of  the 
following  mill  bill  for  a  type  stand.     Allow  25%  for  waste. 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

6  legs 

Chestnut 

2%"  X  zH"  X  43" 

it. 

2  rails 

Chestnut 

W  x  SH"  x  72" 

it- 

2  rails 

Chestnut 

iK"  X  2%"  X  72" 

li. 

3  rails 

Chestnut 

i^"  X  3M"  X  22" 

it 

3  rails 

Chestnut 

I^"   X    2%"   X    22" 

it. 

i  top 

Chestnut 

i"       X  2%"  X  75" 

it- 

Total  

Allowance  

Grand  total.  . 

2.  Find  totals  for  the  following  used  in  making  step  ladders. 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

2 

Chestnut 

H"  x  s5A"  x  90" 

it. 

2 

Chestnut 

H"  X  2>£"  X  75" 

It- 

3 

Chestnut 

K"  x  2ji"  x  36" 

It- 

6 

Chestnut 

H"  X  4H"  X  20" 

it- 

i 

Chestnut 

%"  X  9M"  X  22^" 

it- 

3.  Find  totals  for  the  following  used  in  making  case  racks  for 
print  shop. 

The  waste  in  this  problem  was  due  to  an  error  in  cutting  and 
the  pieces  had  to  be  replaced. 


7o 


INDUSTRIAL   ARITHMETIC 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

24 

Chestnut 

aH"  X  aH"  X  43" 

7i. 

12 

Chestnut 

iK"  X  3M"  X  72" 

7i. 

12 

Chestnut 

iM"  X  2%"  X  72" 

7i. 

18 

Chestnut 

iK"  X  3M"  X  22" 

7i. 

18 

Chestnut 

iK"   X    2^"    X    22" 

7*. 

6 

Chestnut 

W  X  2%"  X  75" 

7t- 

waste  4 

Chestnut 

2%"   X   2%"   X  4l" 

7t- 

i 

Chestnut 

iM"  X  2^"  X  72" 

7t> 

i 

Chestnut 

iM"  X  3M"  X  72" 

7*. 

LESSON  XXXVI 
VOLUME  OF  A  CYLINDER 

1.  In  geometry  it  is  proved  that  the  volume  of  a  cylinder  is 
equal  to  the  area  of  its  base  multiplied  by  its  altitude. 

2.  Find  the  volume  of  each  of  the  following  cylinders: 
Altitude  1 5  ft.,  radius  of  base  7  ft. ;  altitude  10  ft.  radius  of  base 

8  in.;  altitude  18  ft.,  radius  of  base  3  ft.;  altitude  18^  ft., 
radius  of  base  i  ft.;  altitude  25  ft.,  diameter  of  base  zojft.; 
altitude  16  ft.  8  in.,  diameter  of  base  3  ft. 

3.  Find  the  number  of  gallons  a  cylindrical  tank  10  ft.  high 
and  8  ft.  in  diameter  will  hold,     i  cu.  ft.  =  7^  gal. 

4.  At  25 i.  per  cu.  ft.  what  will  be  the  cost  of  a  stone  column 
15  ft.  high  and  2  ft.  in  diameter? 

6.  A  cylindrical  pipe  25  ft.  long  and  6  in.  in  diameter,  inside 
dimensions,  has  water  running  through  it  at  the  rate  of  25  ft. 
per  min.  How  many  gallons  will  pass  through  it  in  i  min.? 
In  i  hr.? 

6.  A  cylinder  4  ft.  in  diameter  and  10  ft.  high  contains  how 
many  times  as  much  volume  as  a  cylinder  2  ft.  in  diameter  and 
10  ft.  high? 

7.  A  prism  4  in.  square  and  2  ft.  long  must  have  how  many 
cubic  inches  cut  from  it  to  give  a  cylinder  4  in.  in  diameter? 

8.  Find  the  volume  of  a  cylinder  the  radius  of  whose  base  is 
R  and  whose  altitude  is  H. 

9.  Use  the  formula  of  problem  8  and  solve  for  H  in  terms 
of  V,  TT  and  R;  for  R  in  terms  of  V,  ir  and  H. 

10.  If  V  =  100  TT,  R  =     5,  find  H. 
If  V  =  250  TT,  H  =  10,  find  R. 
If  V  =  400,      R  =    6,  find  H. 
If  JV  =  500,      H  =  10,  find  R. 

71 


LESSON  XXXVII 
REVIEW 

1.  How  deep  must  a  cylindrical  cistern  6  ft.  in  diameter  be  to 
hold  500  gal.?     1200  gal.? 

2.  A  dealer  was  using  a  cylindrical  measure  6  in.  in  diameter 
and  7%  in.   deep.     He  called  it  a  gallon.     Was  it?    TT  = 
3.1416. 

3.  A  cylinder  whose  altitude  is  h,  radius  of  base  2r,  contains 
how  many  times  as  much  volume  as  a  cylinder  altitude  h  and 
radius  of  base  r? 

4.  A  water  main  is  3  ft.  internal  diameter  and  i  in.  thick. 
Find  the  weight  of  8  ft.     i  cu.  in.  of  pipe  weighs  .26  Ib. 

6.  A  cylindrical  tank  8  ft.  internal  diameter  is  full  of  water; 
if  the  surface  of  the  water  is  lowered  3  ft.,  how  many  gallons 
were  drawn  from  the  tank? 

6.  A  closed  cylindrical  can  8  in.  deep  and  8  in.  in  diameter  is 
placed  inside  of  a  cubical  box  12  in.  each  edge,  inside  diameter. 
How  much  water  can  be  put  into  the  box?    T  =  3^7. 

7.  Which  contains  the  more  volume,  a  cylinder  6  ft.  deep 
and  4  ft.  in  diameter  or  a  cylinder  4  ft.  deep  and  6  ft.  in  diame- 
ter?    The  greater  area? 

8.  How  much  air  passes  into  a  room  through  an  i8-in.  pipe 
each  minute  if  air  is  flowing  through  the  pipe  250  ft.  per  min.? 

9.  A  piece  of  tin  24  in.  long  and  10  in.  wide  is  rolled  into  a 
can;  how  many  gallons  will  the  can  hold  if  ^  in.  is  allowed  for 
the  seams?     (Two  solutions.)     The  top  and  bottom  of  can  do 
not  come  out  of  the  piece  24  in.  X  10  in. 

10.  How  many  loads  of  earth  must  be  removed  to  dig  a  cess- 
pool 10  ft.  deep  and  10  ft.  in  diameter?     i  load  equals  i  cu.  yd. 

11.  A  rotary  air-fan  delivers  2000  cu.  ft.  of  air  per  min. 
through  a  pipe  2  in.  in  diameter.     Find  the  rate  of  discharge  in 
feet  per  minute. 

72 


LESSON  XXXVIII 
REVIEW 

1.  The  following  was  received  for  use  in  the  machine  shops 
of  this  High  School.  Find  its  cost  at  17  f£.  per  Ib.  i  cu.  ft.  of 
steel  weighs  490  Ib.  IT  = 


JESSOP'S  TOOL  STEEL  (ROUND) 


Size 

Total  length 

Weight 

Cost 

.  "t^ii 

I?S 

5 

' 

p 

l%" 

7'5" 

? 

? 

iM" 

20'3" 

? 

? 

iHs" 

i6'S" 

? 

? 

i" 

9'3%" 

? 

? 

X" 

7'7" 

? 

? 

Total. 

2.  Find  the  weight  of  a  piece  of  round  steel  i  ft.  long  and  2 
in.  in  diameter.     Find  its  cost  at  17^.  per  Ib. 

3.  Estimate  the  cost  at  2%£.  per  Ib.  of  the  following  bill  of 
machine  steel  received  at  this  High  School: 

i%  in.  in  diameter  and  36  ft.  long;  i%  in.  in  diameter  and 
37  ft.  long;  \Y±  in.  and  18^  ft.;  i^  in.  and  37^  ft.;  i  in. 
and  8^  ft.;  %  and  26^  ft. 

4.  A  piece  of  cold  rolled  soft  steel  16  ft.  long  and  2  in.  in 
diameter  is  worth  how  much  at  $ff.  per  Ib.? 


73 


LESSON  XXXIX 


FORGE  SHOP  PROBLEMS 

1.  A  shovel  for  the  foundry  was  made  from  the  following 
stock:  %  in.  X  2  in.  X  6^  in.  for  the  blade  and  J^6  in-  round 
X  21  in.  for  the  handle.     Find  the  weight  of  the  shovel,     i  cu. 
ft.  of  iron  weighs  480  Ib. 

2.  Find  the  cost  of  50  shovels  of  problem  i  at  3%i.  Per  Ib. 

3.  A  piece  of  stock  i  in.  X  x  in.  X  15  in.  was  used  to  make 
one  pair  of  tongs.     Find  the  weight  of  35  pairs  of  the  tongs. 


<— 7 


Stoch  y'z  x  l"x  6" 
FIG.  38. 

4.  A  three-way  piece  was  made  from  stock  %  in.  X  i  in.  X 
6  in.     Find  the  cost  of  the  piece  at  3%^.  per  Ib. 

5.  A  clothes  hook  was  made  from  a  piece  of  stock  %  in.  in 
diameter  and  6  in.  long.     Find  the  weight  of  100  of  these  hooks. 

6.  Find  the  weight  of  the  material  used  to  make  a  three-way 
piece  if  the  stock  is  %  in.  in  diameter  and  4^  in.  long. 

74 


FORGE   SHOP   PROBLEMS 


75 


7.  What  is  the  weight  of  100  hooks  for  foundry  shovels  if  the 
stock  is  10  in.  X  ^  in.  in  diameter? 

8.  In  Fig.  38  compute  the  amount  of  stock  in  the  shouldered 
piece  after  3  in.  are  cut  from  each  end. 

1 


i                  :^ 

t 

J 

O'l 

1 

t                              t(N 

LJQ        i 

^                                                 r»   '/     s_ 

t 

r  "                                   l/" 

^             f  I/.       —  •».  rf    ...  -                        9  ^C 

.  s  JT  r*  r 

FIG.  39. 


9.  In  Fig.  39  if  3^2  in.  are  cut  from  each  end  of  piece,  find 
weight  of  shouldered  piece. 


LESSON  XL 
REVIEW 

1.  Show  that  \/3  =  1.732,  also  \/2  =  1.414.     Memorize 
these  numbers. 

?       i2\/2  =  ?       25A/3  =  ?       75\/3  =  ? 

l6\/2     =     ? 


«  +  5^  +  7*6  X  ^ 

3M  +  6>i  +  4   = 


4.  Find  the  volume  of  the  cube  the  area  of  one  of  whose 
faces  is  64  sq.  in. 

5.  How  many  board  feet  in  the  cube  in  problem  4? 

6.  If  a  window  frame  cost  $1.65  what  will  be  the  cost  of  the 
window  frames  in  a  house  with  15  window  openings? 

7.  How  many  board  feet  of  lumber  i  in.  thick  will  be  used  in 
laying  a  floor  28  ft.  X  24  ft.  allowing  25%  for  waste? 

8.  A  man  paid  $35  for  repainting  a  house,  which  was  ]/^Q  of 
the  amount  paid  for  the  property.     Find  the  cost.     If  he  sold 
the  property  for  $1500  did  he  gain  or  lose  and  how  much? 

9.  A  cylindrical  bucket  is  18  in.  in  diameter  and  2  ft.  high. 
How  many  gallons  will  it  hold? 

10.  The  gable  of  a  house  is  12  ft.  high  and  24  ft.  wide.     How 
many  shingles  will  be  required  for  the  two  gables?     Allow  750 
shingles  for  each  100  sq.  ft. 

11.  Vi  57632  =  ?     A/57  -69  =  ?     A/875.  14  =  ? 

12.  An  auger  hole  i  in.  in  diameter  is  bored  through  a  piece 
of  wood  i  ft.  long  and  6  in.  square.     Find  the  volume  remain- 
ing in  the  wood  if  the  hole  is  lengthwise  and  perpendicular  to 
the  base? 

76 


REVIEW 


77 


13.  Find  the  number  of  inches  of  6-in.  round  stock  required 
to  make  a  forging  2)^  in.  in  diameter  and  55  in.  long. 


6 


-86- 


FIG.  40. 


14.  How  long  a  piece  of  6-in.  round  stock  will  be  required 
to  forge  the  above? 


LESSON  XLI 
SIMPLE  EQUATIONS 

Find  the  value  of  the  letter  in  each  of  the  following: 

1.  3x  =  6,  4x  =  16,  5x  =  25,  6x  =  36. 

2.  yx  =  14,  sx  =  30,  77  =  14,  3a  =  15. 

3.  lob  =  30,  i2c  =  36,  1 5k  =  45,  i6p  =  48. 
4-  5P  =  36,  7r  =  15,  8m  =  30,  sL  =  36. 

6.  x2  =  36,  y2  =  25,  m2  =  49,  n2  =  100. 

6.  2x2  =  100,  2x2  =  50,  3n2  =  75,  4s2  =  400. 

7.  6x2  =  216,  5L2  =  80,  4p2  =  64,  iok2  =  6250. 

8.  x2  =  15,  2y2  =  30,  3d2  =  48,  4m2  =  72. 

9.  a  +  5  =  10,  m  +  10  =  25,  y  +  10  =  24. 

10.  r  +  6  =  36,  d  +  7  =  49,  y  +  8  =  50. 

11.  lod  +  4  =  44,  553  +  6  =  31,  76  +  10  =  31. 

12.  8k  +  3  =  27,  lorn  +  5  =  45,  isp  +  10  =  55. 

13.  6x  +  5  =  30,  7y  +  4  =  28,  lor  +  6  =  40. 

14.  x2  +  9  =  25,  x2  +  25  =  169,  y2  +  36    =  100. 

15.  d2  +  36  =  100,  y2  +  36  =  100,  g2  +  100  =  674. 

16.  25  +  a2  =  169,  144  +  d2  =  169,  p2  +  64  =  81. 

17.  If  three  times  a  number  is  increased  by  5,  the  sum  is  14. 
What  is  the  number? 

18.  Seven  times  a  number  and  12  more  is  26.     What  is  the 
number? 

19.  What  number  increased  by  five  times  itself  will  give  30? 

20.  If  John's  money  were  multiplied  by  5  and  $15  added  to 
the  product  the  result  would  be  $100.     How  much  money  has 
he? 

78 


LESSON  XLII 
THEOREM  OF  PYTHAGORAS 

It  is  proved  in  geometry  that 

"  The  square  of  the  hypothenuse  of  a  right  triangle  is  equal  to 
the  sum  of  the  squares  of  its  legs." 


The  legs  of  the  right  triangle  are  usually  represented  by  "a" 
and  "b"  and  the  hypothenuse  by  "c." 
Stated  as  a  formula  the  above  theorem  is 

a2  +  b2  =  c2. 

NOTE. — It  is  also  proved  in  geometry  that  if  the  square  of 
one  side  of  a  triangle  is  equal  to  the  sum  of  the  squares  of  its 
other  two  sides,  the  triangle  is  a  right  triangle,  the  right  angle 
being  opposite  the  greatest  side. 


Exercises 

Find  the  missing  parts  of  the  following  right  triangles: 

79 


80  INDUSTRIAL   ARITHMETIC 

1.  a  =  3,    b  =  4,    c  =  ?  6.  a  =  9,    b  =  ?,    c  =  15. 

2.  a  =  5,    b  =  12,  c  =  ?  6.  a  =  10,  b  =  ?,    c  =  26. 

3.  a  =  ?,    b  =  24,  c  =  25.  7.  a  =  ?,    b  =  15,  c  =  17. 

4.  a  =  18,  b  =  80,  c  =  ?  8.  a  =  ?,    b  =  40,  c  =  41. 

9.  What  is  the  hypothenuse  of  a  right  angle  whose  legs  are 
510  ft.  and  680  ft.  respectively? 

10.  A  house  25  ft.  wide  has  a  gable  10  ft.  high.     Find  the 
length  of  a  rafter. 

11.  Find  the  length  of  the  diagonal  of  a  rectangle  25  ft.  X 
IS  ft. 

12.  Rectangular  frames  are  often  braced  by  a  piece  from  one 
corner  to  the  opposite  lower  corner  of  the  frame.     Find  the 
length  of  a  brace  for  such  a  frame  8  ft.  X  6  ft. 

13.  A  telegraph  pole  30  ft.  high  is  supported  by  a  guy  wire 
fastened  10  ft.  from  its  top  and  anchored  25  ft.  from  its  base. 
What  is  the  length  of  the  guy  wire? 


LESSON  XLIII 
REVIEW 

1.  Find  the  distance  across  corners  of  each  of  the  following 
square  head  screws: 

K  in.;  %  in.;  %  in.;  %  in.;  £{6  in. 

2.  Find  the  distance  across  corners  of  each  of  the  following 
square  bolt  heads: 

\Y±  in.;  2  in.;  4^  in.;  2%  in.;  3%  in. 

3.  A  piece  of  round  stock  2  in.  in  diameter  is  to  be  milled 
square  on  one  end.     What  is  the  side  of  the  largest  square  that 
can  be  cut  from  the  piece? 

4.  Thirty  feet  from  the  mast  head  of  a  derrick  a  boom  is  fas- 
tened to  the  mast.     The  boom  is  25  ft.  long.     One  foot  from 
the  mast  head  is  a  block  containing  two  pulleys  and  ^  ft.  from 
the  end  of  the  boom  is  a  block  with  a  single  pulley.     How  long 
a  rope  must  be  used  to  allow  the  boom  to  stand  at  right  angles 
with  the  mast  and  also  to  have  10  ft.  of  rope  below  point  of 
attachment  of  boom  to  mast? 

5.  The  width  of  a  rough,  U.  S.  standard  nut  or  bolt  head  in 
terms  of  the  diameter  (D)  of  the  bolt  is  W  =  i^D  +  ^  in. 
Find  the  width  of  a  square  nut  for  a  bolt  whose  diameter  is 
i %  in.,  also  the  distance  across  corners. 

6.  If  the  base  of  a  rectangle  is  three  times  its  altitude  and  its 
perimeter  is  40,  find  its  base,  its  altitude,  its  diagonal  and  its 
area. 

7.  The  area  of  a  rectangle  is  288  sq.  ft.     Find  its  base  if  its 
altitude  is  twice  its  base. 

8.  Solve  for  the  letters: 

25  +  3X  =  125,   2Y  +  10  +  5Y  =  79,    2X2  +  125  =  375. 
36  +  X2  =  136,   5Y  +  6Y  -  10  =  84,      40  +  3A2  =  77°. 
6  81 


82  INDUSTRIAL   ARITHMETIC 

9.  Find  the  length  of  the  straight  line  from  the  home  plate 
to  second  base  of  the  baseball  diamond. 

10.  Two  persons  M  and  P  start  from  vertex  A  to  go  to  the 
opposite  vertex  C  of  the  square  ABCD.     M  goes  by  way  of  B 
along  the  perimeter  of  the  square  and  P  on  the  diagonal  AC. 
If  they  both  reach  C  at  the  same  time  what  is  the  ratio  of  their 
speeds,  the  side  of  the  square  being  100? 


LESSON  XLIV 
FACTORING 

1.  The  factors  of  a  number  are  the  numbers  that  multiplied 
together  will  produce  the  number;  e.g.,  the  factors  of  8  are  2,  2 
and  2. 

2.  What  are  the  factors  of   16?    Of  49?    Of  18?    Of  24? 
Of  100?    Of  27? 

3.  Numbers  that  are  produced  by  squaring  a  number  are 
called  perfect  squares;  e.g.,  25  the  square  of  5  is  a  perfect 
square. 

4.  Name  all  the  perfect  squares  between  i  and  144. 

6.  Factor  each  of  the  following  so  that  one  of  the  factors  is 
a  perfect  square: 

75;  So;  125;  150;  18;  27;  45;  63;  8;  12;  20;  24;  98;  147; 
128;  192;  32;  48;  80;  96;  288;  432. 

6.  Find  all  the  factors  of  each  of  the  following: 

150;  275;  95;  100;  240;  3675;  5625;  2468;  357;  7564;  231; 
440;  mi. 

7.  The  product  of  three  numbers  is  720  and  two  of  the  num- 
bers are  8  and  9;  find  the  other  number. 

8.  What  three  numbers  multiplied  together  will  give  144? 
1728? 


LESSON  XLV 
SQUARE  ROOT 

1.  Does    the    \/4  .  9  =  A/4-A/9?    Does    the    \/25  .  4  = 
A/25A/4?     Does    the    A/25  •  J6  =  A/25A/i6?      Does    the 
A/36  .  25  =  \/36\/25? 

2.  These  problems  illustrate  the  following  principle: 

The  square  root  of  the  product  of  two  numbers  is  equal  to  the 
product  of  their  square  roots. 

That  is  \/ab  =  V'a.v'b. 

3.  Find  the  square  root  of  27. 

A/27  =  A/9  •  3  =  A/9  A/3  =  3 A/3  =  3  X  1.732  =  5.196 

4.  Find  the  square  root  of  a2b. 

\/a2  b  =  va*vE  =  aA/b. 
6.  Find  the  square  root  of  each  of  the  following: 
18;  8;  12;  75;  50;  32;  48;  72;  108;  98;  147. 
6.  Simplify: 

Vc2xy;  Vb2d2k;  VpM;  A/a2b2c;  Vm2n2. 


7.  Simplify: 
\/2o;  A/ioo;  \/20 

8.  5  V2  +  3  V2  -  2  A/2  =  ?     8Vs  +  6Vs  ~  3  A/5  =  ? 
7\/3  +  8V3  +  6V3  = 

9\/6  +  3A/6  —  i2\/6  = 


_  =  ?_A/8o  - 

A/75  +  A/27  +  \/48  =  ?     A/24  +  A/ISO  +  A/2i6  =  ? 
\/20o  +  \/8oo  +  \/3200  =  ?     A/432  +  A/363  —  A/3oo  =  ? 

A/3  +  A/8  +  V27  =  ?     Vs  +  V20  +  Vi2  =  ? 

84 


LESSON  XLVI 
REVIEW 

1.  Factor  so  that  one  of  the  factors  shall  be  a  perfect  square  : 
500;  800;  288;  1440;  112;  250;  360;  490. 

2.  Solve  for  the  value  of  the  letter  in  each  of  the  following: 
Sx  +  6x  =  22;  8x  +  10  =  30;  yx  -  5x  =  24;  ;x  +  15  =  50; 

3x  +  5x  +  10  =  14;    4x  +  7X  -  5X  =  255  IOX  +  8x.  +  12  = 
38;  6x  -  sx  +  7x  =  36. 

3.  Simplify: 

\/49°;    A/3a2;    \/2a2;    V$b2; 


4.  If  the  hypothenuse  of  a  right  triangle  is  2  a  and  one  of  the 
legs  is  a,  what  is  the  other  side? 

5.  If  each  leg  of  a  right  triangle  is  m,  what  is  its  hypothenuse? 

6.  Each  side  of  a  square  is  t;  find  its  diagonal. 

7.  If  the  diagonal  of  a  square  is  i5\/2,  what  is  its  side? 

8.  If  the  area  of  a  square  is  144,  find  the  diagonal. 

9.  What  is  the  area  of  a  circle  whose  radius  is  1  2  ? 

10.  The  area  of  a  circle  is  2571-,  what  is  its  radius? 
Solution.—  n-R2  =  2571-;  R2  =  25;  R  =  5. 

11.  Find  the  radius  of  each  of  the  following  circles  whose 
areas  are: 

367r;  4971-;  6471-;  IOOTT;  62571-;  14471-. 

12.  Find  the  diagonal  of  a  square,  whose  area  is  100;  400; 
900;  625;  169;  576. 

13.  What  is  the  area  of  a  square  whose  diagonal  is  8\/2; 
;  3\/2;  i4\/2;  i6\/2;  7\/2j  io\/2;  a\/2;  b\/2  ? 


LESSON  XLVII 
TRIANGLES 

1.  A  triangle  with  two  sides  equal  is  an  isosceles  triangle. 
The  angles  opposite  the  equal  sides  are  called  base  angles  of  the 
isosceles  triangle. 

A  triangle  with  all  its  sides  equal  is  an  equilateral  triangle. 

2.  Construct  from  cardboard  an  isosceles  triangle,  tear  off 
one  of  the  base  angles.    Apply  to  the  other  base  angle.    Are 
they  equal? 

What  statement  can  you  make  about  the  angles  of  an  equi- 
lateral triangle? 

3.  If  one  base  angle  of  an  isosceles  triangle  contains  18°, 
how  many  degrees  in  the  other  base  angle?     If  23°?     If  45°? 
If  60°  ?     If  48°? 

4.  Construct  from  cardboard  a  triangle  with  no  two  sides 
equal.     Cut  off  two  of  the  angles  and  place  them  as  in  the 
following  figure: 


FIG.  42. 

What  kind  of  angle  is  now  formed  at  B  ?    How  many  degrees 
in  it?     What  then  is  the  sum  of  the  angles  of  a  triangle?    The 

sum  of  the  angles  of  a  triangle  is 

86 


TRIANGLES  87 

5.  Find  the  third  angle  of  each  of  the  following  triangles: 
A  =  28°,    B  =  48°,    C  =  ?,  A  =  30°,    B  =60°,    C  =  ?, 

A  =  45°,  B  =  ?,  C  =  90°,  A  =  40°,  B  =  60°,  C  =  ?,     A  = 
60°,  B  =  60°,  C  =  ?,  A  =  100°,  B  =  40°,  C  =  ? 

6.  In  a  right  triangle  what  is  the  sum  of  the  two  acute 
angles? 

7.  If  one  acute  angle  of  a  right  triangle  is  30°,  what  is  the 
other  acute  angle?     If  20°?     If  48°?     If  45°? 

8.  The  vertex  angle  of  an  isosceles  triangle  is  100°.     How 
many  degrees  in  each  base  angle? 

9.  If  a  base  angle  of  an  isosceles  triangle  is  80°,  how  many 
degrees  in  the  vertex  angle? 

10.  In  a  triangle  Z  A  =  2  Z  B  and  Z  C  =  3  Z  B.     How 
many  degrees  in  each  angle  of  the  triangle? 

HINT. — Let  x  =  number  of  degrees  in  ZB. 

11.  In  a  certain  triangle   ZA  =  aZB  and   ZC  =  2ZA. 
Find  the  number  of  degrees  in  each  angle  of  the  triangle. 

12.  Find  the  number  of  degrees  in  the  sum  of  the  angles  of  a 
quadrilateral. 

13.  Construct  a  triangle  with  a  pair  of  angles  equal.    Test 
to  find  out  if  the  sides  opposite  those  angles  are  equal. 

14.  Construct  an  equilateral  triangle.     Draw  its  altitudes. 
Test  to  find  out  if  the  altitude  of  an  equilateral  triangle 
bisects  the  base  to  which  it  is  drawn. 


LESSON  XLVIII 

THE  30°  RIGHT  TRIANGLE 
B 


FIG.  43. 

Theorem.— In  a  30°  right  triangle  the  leg  opposite  the  angle 
of  30°  is  one-half  the  hypoihenuse. 

Given  the  right  triangle  ABC  with  C  the  right  angle,  Z  A  = 
30°,  BC  the  side  opposite  30°  and  AB  the  hypothenuse. 

To  Prove.— BC  =  3^AB. 

Proof.— In  the  ZC  take  ZDCA  =  30°. 

Then  ABAC  is  isosceles  (?)     .'.  AD  =  DC. 

Also  each  angle  of  ADCB  is  60°  (?)     .'.  BD  =  DC. 

/.AD  =  BD,  or  BD  =  3/£AB. 

But  BC  =  BD  (?)     .'.  BC  =  KAB 

Q.  E.  D. 


Exercises 

1.  If  in  the  above  figure  AB  =  100,  120,  200,  300,  400,  18, 
6a,  yb,  4(0  +  d),  8(e  +  f  +  g),  what  is  BC? 

2.  If  in  a  30°  right  triangle  the  side  opposite  30°  is  15}^, 
what  is  the  length  of  the  hypothenuse? 

3.  If  a  kite  string  200  ft.  long  makes  an  angle  of  30°  with  the 
ground,  about  how  high  is  the  kite? 

88 


LESSON  XLIX 
REVIEW 

1.  If  the  hypothenuse  of  a  right  triangle  with  one  of  its 
acute  angles  30°  is  100,  what  is  the  length  of  the  side  oppo- 
site the  angle  of  60°  ? 

2.  A  string  200  ft.  long  attached  to  the  top  of  a  pole  reaches 
the  ground  at  a  point  P.     The  angle  made  by  the  string  and 
a  line  joining  P  with  the  foot  of  the  pole  is  30°.     How  high  is  the 
pole?     How  far  is  P  from  the  foot  of  the  pole? 


Also 


D 

FIG.  44. 

3.  In  this  figure  AB  =  400  ft.     Find  BC,  CD  and  DA. 
find  number  of  degrees  in  /DBA  and  ZBDA. 

4.  The  diagonal  of  a  rectangle  is  30  ft.  and  makes  an  angle 
of  30°  with  the  base.     What  is  the  altitude  of  the  rectangle? 
What  is  the  base?     Its  area? 

5.  A  rope  is  stretched  from  an  upper  corner  of  a  room  15  ft. 
square  and  10  ft.  high  to  the  opposite  lower  corner.     What  is 
the  length  of  the  rope? 

6.  One  rectangle  is  40  ft.  X  20  ft.  and  another  rectangle  is 
80  ft.  X  10  ft.     Find  the  side  of  the  square  that  has  an  area 
equivalent  to  the  sum  of -the  rectangles. 

7.  A  circular  cistern  8  ft.  in  diameter  and  12  ft.  deep  is  full  of 

89 


90  INDUSTRIAL   ARITHMETIC 

water.  If  a  pipe  drains  from  it  10%  of  its  contents  in  i  hr.  and 
another  pipe  conducts  water  to  it  equal  to  8%  of  its  contents  in 
i  hr.,  how  many  gallons  of  water  will  be  in  the  cistern  at  the  end 
of  i  hr.? 

8.  Your  Street  and  Water  Board  placed  157  cylin- 
drical street  markers  at  the  intersections  of  many  principal 
streets.  They  are  9  ft.  high  and  4  in.  in  diameter.  Find  the 
cost  of  the  paint  to  give  them  two  coats,  allowing  i  gal.  to  paint 
250  sq.  ft.  one  coat,  the  paint  costing  $2.50  per  gal. 


Theorem. — In  a  60°  right  triangle  the  side  opposite  60°  is 
one-half  the  hypothenuse  multiplied  by  \/3. 
Use  Fig.  45  and  show  that  x  =  a  \/3' 


FIG.  45- 


6 
FIG.  46. 


Exercises 


1.  If   a  =  10,   c  =  ?,   b  =  ?    If   b  =  8\/3,  c  =  ?,  a  =  ? 
If  c  =  20,  a  =  ?,  b  =  ?    If  a  =  15,  b  =  ?,  c  =  ?    If  b  = 
i8\/3,  c  =  ?,  a  =  ?    If  c  =  40,  a  =  ?,  b  =  ?     (Fig.  46.) 

2.  The  diagonal  of  a  certain  rectangle  makes  an  angle  of  60° 
with  the  base.     If  the  base  is  2o\/3,  what  is  the  diagonal  and 
also  the  area? 

3.  A  ladder  makes  an  angle  of  60°  with  the  ground  and  the 
foot  of  the  ladder  is  i$\/3  ft.  from  the  building  against  which 
it  is  leaning.     How  long  is  the  ladder  and  how  high  does  it 
reach  on  the  wall? 

4.  If  the  width  of  a  house  makes  an  angle  of  60°  with  the 
rafters  and  the  rafters  are  18  ft.  long,  how  high  is  the  gable  and 
how  wide  is  the  house? 

91 


LESSON  LI 
REVIEW 

1.  Fig.  47  represents  a  lathe  center  with  its  dimensions. 
Find  H  and  the  diameter  of  the  small  end  if  the  body  tapers 
.6  in.  per  i  ft. 


^*t«s 
^ 
£ 

<                               r" 

•f-U 

FIG.  47. 


. 

o    •                 —  » 

^            £/ 

0 

/»a 

FIG.  49. 


FIG.  50. 

2.  In  Fig.  48  AB  =  AE  =  CF  =  FG  =  i  in.     If  GE  =  1.3 

in.,  find  the  length  of  the  figure,  also  its  area. 

3.  Find  length  of  AB  also  area  of  Fig.  49. 

4.  When  the  rocking  lever  of  Fig.  50  is  turned  30°  from  the 
horizontal  line  how  much  farther  has  P'  fallen  than  P  has  risen? 

92 


REVIEW  93 


D  G  A 

FIG.  51. 

5.  The  diagram  shows  The  Pratt  truss  used  in  bridge  con- 
struction   DE    and    BA    are    called    struts.     EB   topchord. 
If  the  height  of  the  truss  is  12  ft.;  find  length  of  the  struts, 
topchord  and  length  of  the  bridge. 

6.  An  inclined  plane  20  ft.  long  makes  an  angle  of  30°  with 
the  horizontal  line.     What  is  the  height  of  the  inclined  plane? 

7.  If  in  the  mortar  box  of  Lesson  9,  the  inside  length  of  the 
bottom  is  6  ft.  and  the  inside  height  of  the  sides  is  14  in.,  find 
the  inside  top  length  of  the  box. 


LESSON  LH 
ALTITUDE  AND  AREA  EQUILATERAL  TRIANGLE 

1.  Find  the  altitude  of  the  equilateral  triangle  whose  side  is 
8,  6,  10,  12,  14,  16,  18,  20. 

2.  The  side  of  an  equilateral  triangle  is  17.     What  is  its 
altitude? 

3.  If  the  side  of  an  equilateral  triangle  is  a,  show  that  its 

altitude  is  — \/^.    Learn  this  formula. 

4.  Use  the  result  obtained  in  problem  3  and  give  the  altitude 
if  the  side  is  3,  4,  5,  6,  7,  .87,  13.34,  8,  89.34,  25,  13. 

6.  Find  the  area  of  each  of  the  triangles  in  problem  i  and  2. 

6.  Show  that  the  area  of  the  equilateral  triangle  whose  side 

a2     ,- 
is  a,  is  — V  3-    Learn  this  formula. 

7.  Use  the  result  obtained  in  problem  6  and  give  the  area  of 
each  of  the  following  equilateral  triangles: 

5;  7;  8;  n;  13;  15;  2.5;  3.5;  4.68;  12;  16;  17;  22.2;  38;  40;  50. 

8.  The  base  of  a  prism  is  an  equilateral  triangle  whose  side  is 
5.4.     Find  the  volume  of  the  prism  if  its  altitude  is  6. 

9.  Find  the  volume  of  a  prism  whose  altitude  is  h  and  whose 
base  is  an  equilateral  triangle  side  a. 


94 


LESSON  LIII 


THE  REGULAR  HEXAGON 

If  six  equilateral  triangles  be  ar- 
ranged as  shown  in  Fig.  52  a  regular 
hexagon  is  formed. 

The  point  (o)  that  is  the  common 
vertex  of  the  equilateral  triangles  is 
the  center  of  the  hexagon.  OB  or  OC, 
etc.,  is  a  radius  of  the  hexagon.  Z  A, 
Z  B,  etc.,  are  vertex  angles  of  the 
hexagon. 

Exercises 


FIG.  52. 


1.  How  many  degrees  in  each  vertex  angle  of  a  regular 
hexagon. 

2.  Are  the  sides  of  a  regular  hexagon  equal?     Why? 

3.  Use  problems  i  and  2  to  form  a  definition  of  a  regular 
hexagon? 

4.  What  is  the  perimeter  of  a  regular  hexagon  whose  radius 
is  8,  9,.  10,  a,  b,  x? 

5.  How  many  degrees  in    Z  BOE?      Is   BOE  a  straight 
line?     Why?     In  Z  IOH  if  OH  is  perpendicular  to  CB  and 
OI  is  perpendicular  to  EF,  is  IOH  a  straight  line?     Why? 

In  screws  that  have  hexagonal  heads  and  also  in  hexagonal 
nuts  the  length  of  IOH  is  called  the  distance  across  the  flats,  the 
length  of  BOE  the  distance  across  the  corners. 

6.  Find  the  distance  across  the  flats  of  each  of  the  following 
standard  hexagonal  nuts  having  given  distance  across  the 
corners: 

95 


INDUSTRIAL   ARITHMETIC 


n. 


n.; 


in.; 


n.; 


n. 


n. 


NOTE.  —  Use  \/3  =  *•?  and  express  results  as  multiples  of 

/-     Q 

3^4  in.,  for  example,  the  answer  for  ^  in.  is  -—  which  is  consid- 

ered J'le  in. 

7.  Find  the  distance  across  the  corners  of  each  of  the  follow- 
ing standard  hexagon  nuts  having  given  distance  across  the 
flats: 


in.;        in.; 


in.;  2^  in.; 


n. 


8.  The  distance  across  the  flats  for  both  a  square  and  a  hex- 
agonal bolthead  is  2  in.;  find  the  distance  across  the  corners 
for  each  and  express  the  results  correct  to  three  decimal 
places. 

9.  Find  the  area  of  a  regular  hexagon  whose  side  is  8,  10,  12, 
13-5,  S%'>  a,  b,  f,  m. 

10.  A  regular  hexagon  whose  radius  is  10  is  inscribed  within 
a  circle;  find  area  of  part  of  the  circle  between  the  perimeter 
of  the  hexagon  and  the  circumference. 

11.  Find  the  volume  of  a  prism  whose  base  is  a  regular 
hexagon,  each  side  20.     The  altitude  of  the  prism  is  8.5. 

12.  A  piece  of  steel  2  in.  in  diameter  and  10  in.  long  is  milled 

into  the  largest  possible  hexagonal  piece.     Find  its 
weight  both  before  and  after  being  milled. 

13.  The  distance  across  the  flats  of  the  sleeve 
is  3%  in.  and  is  2%  times  the  diameter  of  the 
hole.  The  hole  is  one-half  the  diameter  of  the 
cylinder  and  one-fourth  the  total  length  of  the 
sleeve.  The  hexagonal  part  is  one-third  the 
total  length.  Find  all  dimensions  and  its  weight 
when  made  of  brass,  i  cu.  ft.  brass  weighs 
524.1  Ib. 


FIG.  53. 


LESSON  LIV 


SCREW  THREADS 

1.  The  depth  of  a  thread  is  the  perpendicular  distance  from 
the  bottom  of  the  groove  to  the  straight  line  joining  the  tops  of 
the  thread.     Twice  this  distance  is  the  double  depth. 

2.  The  root  diameter  of  the  screw  is  the  outside  diameter  of 
the  screw  minus  its  double  depth. 

3.  In  practice  there  are  threads  of  several  shapes.     We  shall 
consider  but  two  kinds,  the  Sharp  V  thread  whose  angle  is  60° 
and  the  U.  S.  standard  form  of  thread. 

!«—  P— J 


Vvl 

U.S.  Standard  Thread 
FIG.  54- 


60°  V  Thread 
FIG.  55. 


The  U.  S.  standard  has  the  same  angle  as  the  60°  thread,  but 
has  its  top  and  its  bottom  flat.  The  width  of  this  flat  is  one- 
eighth  of  the  pitch  of  the  thread.  The  depth  of  this  thread  is 
three-fourths  of  the  depth  of  the  60°  V  thread  of  the  same 
pitch. 

r      EXERCISES 
(These  problems  refer  to  Screws  of  Single  Thread) 

1.  A  screw  has  15  threads  per  in.     What  is  its  pitch?     Its 
lead. 

7  97 


98  INDUSTRIAL   ARITHMETIC 

2.  Find  the  pitch  and  lead  for  the  following  number  of 
threads  per  inch: 

13;  14;  25;  36;  8;  17;  18;  20. 

3.  Determine  the  depth  of  each  of  the  following  60°  V 
threads  whose  pitches  are  H2  in-,  He  in-,  Ho  in.,  %  in., 
Y±  in.,  K  in.,  K  in. 

4.  Show  that  the  depth  of  the  60°  V  thread  is  equal  to  its 
pitch  multiplied  by  3^  ^3  or  about  .866. 

6.  Find  the  root  diamters  of  each  of  the  screws  in  problem  3 
if  the  outside  diameter  of  each  is  2  in. 

6.  Determine  the  depth  of  thread  for  the  pitches  given 
in  problem  3  for  the  U.  S.  standard-shaped  threads. 

7.  How  many  revolutions  will  a  60°  V  thread  make  in 
advancing  2  in.,  if  its  pitch  is  ^{Q  in.? 

8.  If  the  flat  of  a  U.  S.  standard-shaped  thread  is  %  in., 
what  is  its  depth? 

9.  If  the  root  diameter  of  a  60°  V  thread  with  10  threads  per 
in.  is  i Y±  in.,  what  is  its  outside  diameter? 

10.  Find  the  outside  diameter  of  the  U.  S.  standard  with  8 
threads  per  in.  and  a  root  diameter  of  i%  in. 


LESSON  LV 
REVIEW 

1.  How  many  revolutions  will  a  60°  V  thread  make  in  ad- 
vancing \Y±  in.  if  its  pitch  is  3^o  in-? 

2.  What  is  the  depth  of  the  thread  in  problem  i? 

3.  If  the  flat  of  a  U.  S.  standard-shaped  thread  is  ^fg  m-> 
what  is  its  depth? 

4.  The  root  diameter  of  a  60°  V  thread  with  12  threads  to 
the  inch  is  %  in.     What  is  its  outside  diameter? 

5.  The  outside  diameter  of  a  U.  S.  standard  is  i  in.  and  its 
pitch  34  in.     If  the  screw  is  2  in.  long  what  is  the  length  of  the 
thread? 

6.  If  the  outside  diameter  of  a  sharp  V  thread  is  ^  in.  and 
its  pitch  3"{o  m-  what  is  the  length  of  the  thread  of  a  screw  3  in. 
Long? 

7.  If  the  feed  is  ^2  m-  and  the  cutting  speed  20  ft.  per  min., 
find  the  required  time  for  the  tool  to  advance  2  in.  along  a  piece 
3  in.  in  diameter. 

8.  When  cutting  hard  steel  the  speed  may  be  33  ft.  per  min. 
if  the  depth  of  the  cut  is  34  m-  and  the  feed  3^6  m-     Find  the 
number  of  R.P.M.  a  cylinder  of  hard  steel  2  in.  in  diameter  is 
making  for  the  above  data. 

9.  If  the  scale  is  %  in.  =  5  ft.  o  in.  what  is  the  length  on  a 
working  drawing  for  each  of  the  following: 

2  ft.  6  in.;  4  ft.;  10  ft.;  12  ft.  8  in.;  25  ft.;  48  ft.? 
If  the  scale  is  34  in.  =  3  ft.  o  in.?     i  in.  =  5  ft.  6  in.? 
%  in.  =»  4  ft.  6  in.? 

10.  A  carpenter  has  a  mitre  box  5  in.  wide  outside  dimen- 
sions.    Explain  how  he  will  find  the  points  on  the  box  through 
which  he  must  saw  to  get  a  45°  angle;  a  60°  angle. 

99 


100 

11. 


INDUSTRIAL   ARITHMETIC 


FIG.  56. 

The  above  figure  is  a  side  view  of  an  oblique  thrust.  R, 
P"  and  P  are  in  a  straight  line.  Find  area  of  shaded  portion, 
length  of  SP  and  S'P'. 

12.  Find  the  nearest  number  of  64ths  to  which  each  of  the 
following  fractions  is  equivalent: 

15     ii     6      9      21     125     ii     19     13 
17  '  13  '  7  '  ii  '  23'   127'  15  '  21  '  14 

13.  State  the  limit  of  the  error  for  each  of  your  answers 
in  problem  12. 

14.  If  R  =  5  and  r  =  3,  find  value  of  V  in 

V  =  -v(R*-r*). 

O 

15.  If  TI  =  6,  r2  =  3,  and  h  =  2.4  find  value  of  V  in 


16.  Use  the  following  formula 


86,670 


SCREW   THREAD 
to  find  the  value  of  T  in  the  table 


101 


r 

D 

5 

P 

r 

D 

5 

P 

7 

5 

100 

18 

5 

1  80 

7 

5 

200 

10 

5 

1  20 

8 

5 

1  60 

16 

5 

140 

12 

5 

2220 

9 

5 

200 

17.  How  many  degrees  in  each  angle  of  the  triangle  ABC 
if  ^A  =  3/5  and  ZC  =  2 

18.  Simply: 

\/io8  —  \27;  \5o       \2oo; 


A/3°o  - 
19.  If  the  feed  is 


+       i2;  A/32  +  X/48- 
in.  and  the  work  has  a  speed  of  30 


R.P.M..  how  long  will  be  required  to  cut  a  piece  i  foot  long? 

20.  Floor  moulding  is  required  for  a  floor  36'  X  24'.     If 
the  strips  can  be  purchased  10',  12'  or  16'  long,  which  length 
would  you  purchase  assuming  no  waste  and  the  price  the  same 
per  foot?     Why? 

21.  What  is  the  side  of  an  equilateral  triangle  whose  area 
is  25\/3?     2o\/3?     ioo\/3? 

22.  Solve  for  the  letter  in  each  of  the  following: 

5-y2  +  io=no;  25  +  a2  =  75;  15  +  62  =  115. 
5/>2  +  7  =  232;  36  +  k2  =  436;  29  +  d2  =  173. 
3*2  +  6  =  426;  10  +  e2  =  210;  17  +  g*  =  417. 


LESSON  LVI 
THREAD  CUTTING 

Mathematically,  thread  cutting  is  merely  a  problem  of  gears. 

The  cutting  tool  is  made  to  advance  along  the  piece  being  cut 
by  means  of  a  lead  screw.  The  piece  is  made,  to  revolve  as  the 
tool  advances.  If  the  number  of  R.P.M.  of  the  piece  is  the 
same  as  the  R.P.M.  of  the  lead  screw,  the  tool  will  evidently 
cut  on  the  piece  the  same  number  of  threads  per  inch  as  the 
number  on  the  lead  screw;  e.g.,  if  the  lead  screw  has  4  threads  per 
in.  the  tool  will  cut  4  threads  per  in.  on  the  piece.  If  the 
R.P.M.  of  the  piece  is  twice  the  R.P.M.  of  the  lead  screw,  then 
8  threads  per  in.  will  be  cut. 

The  diagram  (Fig.  57)  of  the  head  stock  of  a  simple  geared 
lathe  will  help  to  make  clear  how  the  number  of  R.P.M.  of  the 
work  and  of  the  lead  screw  is  accomplished. 

I,  I'  and  I"  are  intermediates,  hence  have  no  effect  on  R.P.M. 
I  and  I'  determine  the  direction  of  rotation  of  the  lead  screw. 
C  and  S'  are  both  keyed  to  the  stud  shaft.  S'  and  S  in  the 
problems  that  follow  are  equal.  C  and  L  are  the  gears  that  by 
changing  determine  the  R.P.M.  of  the  lead  screw. 

Standard  "change  gears  "are  made  in  series  that  have  a  com- 
mon difference  or  either  4  teeth  or  5  teeth.  The  smallest  gear 
of  each  series  contains  20  teeth,  and  the  largest  1 20  teeth  and 
100  teeth  respectively. 

20,  24,  28,  32,  36,40, 44, 48,  52,  56,  60,  .    .   120,  .    .  A  series. 
20,  25, 30,  35, 40, 45,  50,  55,  60,  65,  70,  .    .   100,  .    .  B  series. 

Lathes  are  also  supplied  with  other  gears  as  46,  66,  69. 

102 


THREAD   CUTTING 


103 


Idler      I 


Fixed  Gear  S' 
on  Stud 


S    Gear  on  Spindle 
Carries  Work 


I.'    Idler 


6  Change  Gear 
on  Stud 

I3*  Intermediate 


FIG.  57- 


104  INDUSTRIAL   ARITHMETIC 

Exercises 

1.  In  the  above  diagram  S  is  made  to  revolve;  explain  how 
the  motion  is  communicated  to  L. 

2.  S  =  S'.     If  the  number  of  teeth  of  C  equals  twice  the 
number  of  teeth  of  L  which  is  revolving  the  faster  and  how 
many  times  as  fast?     If  L  has  6  threads  per  in.,  how  many 
would  be  cut  on  the  work? 

3.  If  C  =  L  and  S  =  ^S',  compare  number  of  R.P.M.  of  C 
with  that  of  L. 

4.  If  C  =  YJL  and  S  =  ^S'  compare  R.P.M.  of  C  with  that 
ofL. 

5.  If  C  =  L,  S  =  S'  and  I"  =  3L,  compare  R.P.M.  of  C 
with  that  of  L. 

6.  If  the  lead  screw  has  8  threads  per  in.  how  many  threads 
per  inch  are  being  cut  on  a  piece  making  twice  as  many  R.P.M.  ? 
3  times  as  many?     2%  times  as  many?     How  many  times  as 
many  to  cut  20  threads  per  in.?     30  threads  per  in.?     40  threads 
per  in.?     12  threads  per  in.? 

7.  The  lead  screw  has  4  threads  per  in.,  and  L  has  80  teeth. 
How  many  teeth  must  C  have  in  order  that  8  threads  per  in. 
may  be  cut?     16  threads  per  in.? 

8.  What  gears  on  C  and  L  from  the  A  series  may  you  use  to 
cut  12  threads  per  in.  with  a  lead  screw  of  6  threads  per  in.? 
5  threads  per  in.?     8  threads  per  in.? 

9.  The  lead  screw  of  a  lathe  has  5  threads  per  in.     What 
gears  on  C  and  L  may  you  use  to  cut  10  threads  per  in.?     12 
threads  per  in.?     15  threads  per  in.?     3  threads  per  in.?     18 
threads  per  in.? 

10.  With  a  lead  screw  of  5  threads  per  in.,  what  is  the  largest 
number  of  threads  per  inch  that  can  be  cut  in  a  simple-geared 
lathe  withS  =  S'? 

11.  The  standard  pipe  thread  for  i^-in.  pipe  is  n^  threads 
per  in.     What  gear  will  you  use  to  cut  this  thread  with  a  lead 


THREAD   CUTTING  105 

screw  of  6  threads  per  in.?     5  threads  per  in.?     4  threads  per 
in.? 

12.  Let  NT  =  No.  of  teeth  of  L,  NR  =  R.P.M.  of  L,  N'T  = 
No.  of  teeth  of  S,  N'R  =  R.P.M.  of  S.    Then  show  that 
NT        /N'R\      Threads  per  inch  to  be  cut 


N'T       VNR  /  T  Threads  of  L 

13.  Use  the  formula  of  problem  12  to  find  number  of  threads 
that  will  be  cut  per  inch  when  N'T  =  30,  NT  =  20  andL  =  6; 

N'T  =  40,  NT  =  45,  L  =  4;  NT  =  65,  N'T  =  55,  L  =  8;  NT 
=  120,  N'T  =  48;  L  =  5;  NT  =  ioo,  N'T  =  85,  L  =  8;  L  = 
6,  NT  =  60,  N'T  =  30;  N'T  =  85,  NT  =  80,  L  =  8;  L  =  6, 

NT  =  52>  N'T  =  40- 

Very  often  the  number  of  teeth  of  S  and  of  S'  are  not  equal, 
the  number  of  S  being  less  than  the  number  of  S'.  The  effect  of 
this  is  to  make  S  revolve  more  rapidly  than  S'  and  therefore 
than  L  provided  C  and  L  are  equal.  Suppose  S  =  20  and  S'  = 
40;  then  S  will  make  two  revolutions  while  S'  and  L  are  making 
one.  Hence,  if  L  has  6  threads  per  in.,  12  threads  will  be  cut 
on  the  work.  That  is,  we  get  the  same  result  with  S  =  20, 
S'  =  40  and  L  =  6,  as  if  we  had  S  =  S'  and  L  =  12.  This 
illustrates  the  following:  When  S  and  S'  are  not  equal,  we 
assume  C  and  L  to  be  equal,  find  the  number  of  threads  the  given 
lead  screw  will  cut  per  inch,  use  this  number  as  the  lead  screw  and 
proceed  as  in  the  previous  work.  This  number  is  called  the  lead 
number  or  the  calculating  lead  screw. 

Example. — If  S  =  30;  S'  =  40,  and  L  =  6,  what  gears  may 
be  used  to  cut  12  threads  per  in.? 

Solution. — Since  S/S'  =  %,  S  will  make  i^  revolutions, 
while  L  is  making  one,  that  is,  6  X  1^3  or  8  threads  per  in.  will 
be  cut.  Hence  the  lead  number  is  8.  Also  x%  =  4%2J 
therefore  gears  of  48  and  32  must  be  used,  32  on  C  and  48  on  L. 

Stated  as  a  formula  the  above  is 

Threads  per  inch  to  be  cut  _  NT  of  L 
Lead  number  NT  of  C 


io6 


INDUSTRIAL  ARITHMETIC 


Many  lathes  are  made  with  the  number  of  teeth  of  S  and  of 
S',  30  and  40  respectively. 

Exercises 

1.  If  S  =  30,  S'  =  40,  find  the  lead  number  when  L  =  4,  L 
=  6,  L  =  8,  L  =  5. 

2.  If  S  =  30,  S'  =  40  and  L  =  6,  what  gears  may  be  used  to 
cut  10  threads  per  in.?     15  threads  per  in.?     16  threads  per 
in.?     20  threads  per  in.?     4  threads  per  in.?    40  threads  per 
in.? 

3.  If  S  =  30,  S'  =  40  and  L  =  6,  what  gears  may  be  used  to 
cut  a  standard  pipe  thread  for  %-in.  pipe? 

4.  With  S  =  30,  S'  =  40,  L  =  6  and  the  A  series,  what  is  the 
largest  number  of  threads  per  inch  that  can  be  cut  in  a  simple- 
geared  lathe?     The  least  number? 


FIG.  58. 

In  the  above  train  of  gears  B  and  C  are  keyed  to  the  same 
shaft,  also  D  and  E.  Such  a  train  is  said  to  be  compounded. 
If  the  train  is  set  in  motion  by  A  or  A  drives  C,  then  A  is  called 
the  driving  gear  of  the  train  and  F  the  driven  gear.  Also 
A,  B,  D  are  called  the  drivers  and  C,  E,  F  the  driven. 

Law  for  Compound  Gears.— 

NA  =  R.P.M.  of  A  and  NF  =  R.P.M.  of  F 
A,  B,  C,  etc.  =  the  number  of  teeth  of  each  gear. 


THREAD   CUTTING  IOJ 

Then 

NA  _  C.E.F 

NP  ~  A.B.D' 

Expressed  in  words  this  formula  is:  The  R.P.M.  of  the  driving 
gear  divided  by  the  R.P.M.  of  the  driven  gear  of  a  train  is  equal  to 
the  product  of  the  number  of  teeth  of  the  driven  gears  divided  by 
the  product  of  the  number  of  teeth  of  the  driving  gears. 

Exercises 

1.  If  NA  =  10,   A  =  20,   B  =  18,  C  =  30,  D  =  22,  E  = 
26  and  F  =  40.     Kind  NF. 

2.  Solve  problem  i  without  the  use  of  the  formula. 

3.  Find  NA  when  A  =  10,  C  =  20,  B  =  15,  E  =  30,  D  = 
18,  F  =  36  and  NP=  20. 

4.  Find  NF  when  NA  =  8,  A  =  12,  C  =  24,  B  =  18,  E  = 
36,  D  =  20  and  F  =  40. 

Compound-geared  Lathe. — A  lathe  is  said  to  be  compound 
geared  when  there  are  two  changes  of  speed  between  S',  the 
fixed  gear  of  the  studshaft,  and  L,  'the  gear  of  the  lead 
screw.  Fig.  59  shows  one  way  of  compounding  the  gears  of 
a  lathe. 

C'  is  another  gear  keyed  to  the  same  shaft  with  I".  The 
change  gears  are  L,  C',  I"  and  C.  I"  and  C'  are  together 
known  as  the  compound  and  one  gear  is  usually  twice  the 
other,  as  60  and  30.  Compound  gearing  is  used  when  in  a 
simple-geared  lathe  a  gear  with  a  larger  number  of  teeth  than 
is  usually  made  would  be  needed  or  a  gear  too  large  for  the 
center  distance  between  the  shafts. 

Law  for  Compound-geared  Lathe. — 

Threads  to  be  cut  per  inch  _  I."L  _  I"      L 
Lead  number  =  C'.C  =Cr.XC 

The  following  will  illustrate  how  to  apply  this  formula. 


io8 


INDUSTRIAL   ARITHMETIC 


FIG.  59. 


Find  the  gears  necessary  to  cut  n3^  threads  per  in.  with 
a  lead  screw  of  6  threads  per  in.  on  a  lathe  with  S  =  30 
and  S'  =  40. 


Lead    number  =  8;  then-^   = 

o 


X  %  = 


X  6%o-     That  is  7,7   =    — and  ~  =  \  -,  the  required 
C          30          L       64 


gears. 


Since  I"  and  C'  are  usually  60  and  30,  we  should  try  to  make 


COMPOUND-GEARED    LATHE  IOQ 

the  fraction  equal  to  the  product  of  two  fractions,  one  of  which 
is  either  ^  or  %. 

Thread  Cutting 

1.  With  S  =  S';  find  the  compound  gearing  necessary  to 
cut  32  threads  per  in.,  with  a  lead  screw  of  /^-in.  pitch. 

2.  With  S  =  30  and  S'  =  40,  find  the  compound  gearing  that 
may  be  used  to  cut  22  threads  per  in.  with  a  lead  screw  of 
pitch  %  in. 

3.  With  S  =  30,  S'  =  40  and  L  =  ^-in.  pitch,  what  simple 
gearing  will  cut  23  threads  per  in.?     What  compound  gearing? 

4.  Make  L  =  8  threads  per  in.  in  problem  3  and  then 
solve  it. 

6.  Make  L  =  6  threads  per  in.  in  problem  3  and  cut  27 
threads  per  in. 

6.  With  S  =  S'  andL  =3^-in.  pitch,  find  gears  for  cutting 
24  threads  per  in.  by  compounding. 

7.  Make  S  =  30  and  S'  =  40  in  problem  6  and  solve  it. 


LESSON  LVII 
TAPER 

A  piece  of  turned  work  with  uniformly  increasing  diameter 
is  called  a  taper. 

The  difference  of  the  diameters  of  the  two  ends  of  the  taper  is 
the  taper  of  the  piece.  It  is  usually  given  as  a  certain  number 
of  inches  per  foot  or  per  inch.  For  example,  if  a  taper  is  i  ft. 
long  with  diameters  of  2  in.  and  i  in.  respectively,  the  taper  is 

1  in.  per  ft.  or  ^{2  m-  Per  m- 

Exercises 

1.  If  the  larger  diameter  of  a  taper  18  in.  long  is  3  in.  and  the 
taper  %  in.  per  in.,  what  is  the  smaller  diameter? 

2.  The  diameters  of  a  taper  are  2  in.  and  i  in.     If  the  taper  is 
^ 6  m-  Per  in->  find  the  length  of  the  taper. 

3.  What  is  the  difference  in  the  diameters  of  a  taper  10  in. 
long  whose  taper  is  %  in.  per  ft.? 

4.  What  is  the  taper  per  foot  of  a  taper,  3^  in.  with  di- 
ameters %  in.  and  J^o  in.? 

6.  The  taper  per  foot  of  a  Jarno  taper  is  .6  in.  Find  the 
length  of  a  Jarno  taper  whose  diameters  are  %  in.  and  %Q  m- ', 

2  in.  and  1.6  in.;  i  in.  and  .8  in. 

6.  The  length  of  a  Jarno  taper  is  10  in.  and  its  larger  diame- 
ter 2.5  in.;  find  its  smaller  diameter.     If  the  length  is  g%  in. 
and  the  smaller  diameter  1.9  in.,  what  is  the  larger  diameter? 

7.  The  taper  per  foot  of  a  Brown  and  Sharp  taper  is  %  in. 
Find  the  length  if  the  two  diameters  are  2.25  in.  and  2.58  in. 


TAPER  III 

If  the  length  is  7%  in.  and  the  larger  diameter  2.052  in.,  find 
the  smaller  diameter. 

8.  The  diameters  of  a  taper  are  1.045  m-  and  1-348  in.  and 
the  taper  per  foot  .516  in.;  find  the  length  of  the  taper. 

9.  What  is  the  taper  per  foot  of  a  piece  9^6  m-  long  with 
diameters  of  i%  in.  and  2^  in.? 

10.  If  the  length  of  a  taper  is  L  ft.,  and  the  diameters  of  its 
ends  a  in.  and  b  in.,  what  is  its  taper  (T)  per  foot? 

11.  Use  the  formula  to  find  T  when  L  =  2,  A  =  i,  B  = 
2;  when  L  =  6  in.,  A  =  i,  and  B  =  144;  when  L  =  10  in., 
A  =  2,  B  =  3.375;  when  T  =  .623,  A  =  1.02  and  B  =  1.28. 
Find  L. 


LESSON  LVIII 
TAPER  TURNING 

Tapers  are  turned  in  a  lathe  either  by  means  of  the  taper 
attachment  or  by  offsetting  the  tailstock.  When  the  taper  at- 
tachment is  used  the  taper  in  inches  per  foot  is  determined  and 
the  taper  attachment  set  to  that  number.  If  the  tailstock  is 
offset  we  must  know  the  taper  per  foot  and  the  length  of  the 
piece  expressed  in  feet. 

Then  if  S  =  offset  in  inches,  T  =  taper  in  inches  per  foot 
and  L  =  length  of  piece  in  feet,  we  have  as  the  formula  for  the 
offset,  or 


NOTE.  —  The  proof  of  this  formula  depends  upon  principles  of 
geometry. 

Exercises 

1.  A  cylinder  i  ft.  long  is  to  be  tapered  Y±  in.  per  ft.     How 
much  must  the  tailstock  be  offset? 

2.  Determine  the  offset  for  each  of  the  following: 

Taper  %  in.  per  ft.,  piece  8  in.  long;  taper  3^  in.  per  ft.,  piece 
18  in.  long;  taper  .6  in.  per  ft.,  piece  10  in.  long;  taper  .602  in. 
per  ft.,  piece  14  in.  long;  taper  .625  in.  per  ft.,  piece  5  in.  long; 
taper  .592  in.  per  ft.,  piece  25  in.  long. 

3.  The  tailstock  of  a  lathe  is  offset  2  in.     Find  the  taper  if 
the  piece  is  8  in.  long;  10  in.  long;  2  ft.  long. 

4.  If  the  offset  is  %  in.  and  the  taper  %  in.  per  ft.,  what  is 
the  length  of  the  piece? 


TAPER   TURNING 


5. 


How  much  must  the  tailstock  be  offset  to  turn  the  piece 
shown  in  the  figure? 


^t 

Sf 

7 

3 

,                    7"                  » 

4"            > 

,                               TI" 

II 

FIG.  60. 

6.  A  lathe  center  5  mi  long  is  to  be  tapered  .6  in.  per  it. 
Find  the  offset. 

7.  A  taper  pin  4^  in.  long  has  the  large  end  .49  in.  in  diame- 
ter and  the  small  end  .398  in.  in  diameter.     How  much  was  the 
tailstock  offset  to  turn  the  piece? 


FIG.  61. 

Determine  each  offset  for  turning  the  piece  shown. 

9.  Seller's  taper  is  %  in.  per  ft.     Determine  the  length  of  a 
piece  being  turned  to  this  taper  if  the  set  over  is  %  in. 

10.  The  Jarno  taper  No.  18  is  9  in.  long  and  has  diameters  of 
1.8  in.  and  2%  in.     What  must  be  the  offset  to  turn  this 
taper? 

11.  How  far  must  the  tailstock  be  set  over  to  taper  a  piece 
15  in.  long  with  American  taper?     (A.  T.  =  %e  m-  Per  ft.) 


LESSON  LIX 
REVIEW 

1.  Find  the  altitude  of  the  equilateral  triangle  whose  side  is 
3^  in.,  %  in.,  %  in.,  correct  to  three  decimal  places. 

2.  A  boiler  tube  is  to  be  made  3  in.  in  inside  diameter,  ^f  e  in- 
thick  and  10  ft.  long.     How  long  a  piece  of  brass  3  in.  square 
will  be  required  to  make  the  tube? 

3.  Find  the  length  of  the  line  ABCD  as  in  the  figure. 


A            B^H  *$  i 

tr                                        in' 

10 

FIG.  62. 

4.  Find   the  altitude  of  each  of  the  equilateral  triangles 
of  Fig.  63. 

5.  In  turning  a  locomotive  wheel  78  in. 
in  diameter,  what  is  the  proper  number 

of  revolutions  per  minute,  in  order  that 

FIG.  63. 
the  cutting  speed  may  be  10  ft.  per  mm.? 

6.  A  piece  of  brass  4  in.  in  diameter  is  making  80  R.P.M- 
What  is  the  speed  of  a  point  on  its  surface? 

7.  If  a  piece  tapers  .0026  in.  per  in.,  what  is  its  taper  per  foot? 

8.  The  standard  pipe  thread  taper  is  %  in.  per  ft.     How 
much  must  the  tailstock  be  offset  to  turn  this  taper  on  a  piece 
2  ft.  long? 

9.  A  round  shaft  is  3'^  in.  in  diameter.     Find  the  length  of 
the  greatest  square  end  that  can  be  made  on  the  shaft. 

114 


REVIEW  115 

10.  If  S  =  30  and  S'  =  40  and  L  =  3^-in.  pitch,  find  the 
simple  gearing  you  may  use  to  cut  5  threads  per  in. ;  also  the 
compound  gearing. 

11.  If  the  scale  is  i"  =  o'  8"  what  should  be  the  length 
of  each  line  for  a  scale  drawing  of  problem  3. 

12.  What  is  the  micrometer  reading  for  each  of  the  following: 
.777     in.?     .326    in.?      .565     in.?     .480    in.?     .444    in.? 

•345  in.? 

13.  What  is  the  nearest  number  of  64ths  for  each  number  of 
problem  12? 

14.  The  diameter  of  a  drill  is  2  in.     Its  speed  is  92  R.P.M. 
and  its  feed  per  revolution  is   .015  in.    How  many  cubic 
inches  are  being  removed  per  minute? 

15.  A  drill  4  in.  in  diameter  making  46  R.P.M.  removes  10.8 
cu.  in.  per  min.     Find  its  feed  per  revolution. 


LESSON  LX 
RATIO 

The  quotient  obtained  by  dividing  a  by  b  is  called  the  ratio 
of  a  to  b.     The  ratio  of  a  to  b  is  written  as  a:b,  or  a/b  or  a-f-b. 

Exercises 

1.  Find  the  value  of  each  of  the  following  ratios: 

10:5;  16:20;  32:8;   100-^25;  17-5-19;  25-5-5;  !%4;  34i?; 
172^44;  a2/a;  b4/b2;  c3/c2;  d«/d3. 

2.  Find  the  ratio  of  the  areas  of  two  rectangles  of  altitudes 
5  and  10  and  bases  8  and  16  respectively. 

3.  What  is  the  ratio  of  the  areas  of  two  triangles  of  altitudes 
10  and  1 8  and  bases  22  and  30  respectively? 

4.  The  radii  of  two  circles  are  id  ft.  and  15  ft.     What  is 
the  ratio  of  their  areas?     Of  their  circumferences? 

6.  Two  gears  of  60  and  45  teeth  respectively.     What  is  the 
ratio  of  their  speeds?     If  they  have  a  and  b  teeth? 

6.  Find  the  ratio  of  the  speeds  of  two  pulleys  connected  by  a 
belt  if  their  diameters  are  12  in.  and  8  in.;  15  in.  and  7  in.;  a 
and  b. 

7.  What  is  the  ratio  of  the  areas  of  two  circles  whose  radii 
are  p  and  q? 

8.  What  is  the  ratio  of  the  surface  speeds  of  the  pulleys  in 
problem  6? 

9.  The  ratio  of  the  speeds  of  the  driving  gear  to  the  driven 
gear  is  3:2.     If  the  driving  gear  containd  48  teeth,  how  many 
teeth  has  the  driven  gear? 

116 


RATIO 


117 


10.  The  efficiency  of  a  machine  is  the  ratio  of  the  units  of  work 
given  out  by  the  machine  and  utilized  to  the  total  number  of 
units  of  work  put  into  it.  What  is  the  efficiency  of  a  machine 
that  gives  out  37  ft.-lb.  from  50  ft.-lb.  put  into  it?  20  ft.-lb. 
from  45  ft.-lb.?  38  ft.-lb.  from  48  ft.-lb.? 


FIG.  64. 

11.  The  ratio  a/ioo  is  called  the  grade  of  the  slope.     The 
loo  ft.  is  horizontal  distance  and  a  the  vertical. 

If  a  road  bed  rises  10  ft.  each  100  ft.  measured  horizontally, 
what  is  its  grade?     What  is  a  7%  grade?     10%  grade? 

12.  A  vertical  rise  of  20  ft.  in  1000  ft.  is  what  grade? 

13.  Water  consists  of  2  parts  hydrogen  and  i  part  oxygen. 
What  is  the  ratio  of  oxygen  to  hydrogen?     Of  hydrogen  to 
oxygen?     Of  hydrogen  to  water?     Oxygen  to  water? 

14.  Name  several  gears  that  will  have  a  speed  ratio  of  5:2. 

15.  The  volumes  of  two  sphere's  have  the  same  ratio  as  the 
cubes  of  their  radii.     Find  the  ratio  of  the  volumes  of  two 
spheres  whose  radii  are  2  and  3;  4  and  5;  and  7  and  9;  10  and  12. 

16.  Find  the  ratio  of  the  areas  of  two  squares  whose  sides  are 
m  and  n  respectively. 

17.  What  is  the  ratio  of  the  lead  to  the  pitch  of  a  single- 
thread  screw? 

18.  If  the  scale  is  %  in.  =  2  ft.  o  in.,  what  is  the  ratio  of  a 
scale  drawing  2  in.  long  to  the  length  of  the  line  it  represents? 

19.  A  speed  ratio  of  7:3  is  required  for  two  gears.     If  the 
driven  gear  has  56  teeth  how  many  must  the  driving  gear  have? 

20.  If  pigiron  contains  93%  pure  iron,  3%  carbon  and  2% 
sulphur,  find  the  ratios  of  the  different  elements  given. 


Il8  INDUSTRIAL   ARITHMETIC 

21.  The  heating  surface  of  a  certain  boiler  is  1800  sq.  ft. 
The  grate  measures  9  ft.  X  8  ft.    Find   the  ratio  of  grate 
surface  to  heating  surface. 

22.  If  the  ratio  of  two  numbers  is  9 :  7  and  one  of  the  numbers 
is  1423,  what  is  the  other  number?     Two  solutions. 


LESSON  LXI 
SECTORS  AND  SEGMENTS 

The  part  of  a  circle  between  two  radii  and  an  arc  is  called  a 
sector  of  the  circle,  as  AOB. 

The  part  of  a  circle  between  an  arc  and 
a  chord  is  called  a  segment  of  the  circle, 
the  shaded  part  of  Fig.  65. 

The   angle   at   the   center   of  the  circle 
formed  by  the  two  radii  is  called  the  angle 
_  of  the  sector. 

A  "^**LI  |    ,  L  ,    -J->^  O 

Exercises 

1.  What  part  of  the  whole  circle  is  a  sec- 
tor whose  angle  is  6o°?  120°?  30°?  90°?  180°?  45°?  15°?  38°? 

2.  Find  the  area  of  a  sector  of  90°,  if  the  radius  of  the  circle  is 
10,  8,  12,  5,  25,  15,  35. 

3.  If  the  radius  of  a  circle  is  10,  find  the  area  of  a  sector  of 
60°,  90°,  45°,  30°,  180°,  36°,  18°. 

4.  If  the  area  of  a  sector  AOB  is  28  and  the  triangle  AOB  is 
20,  what  is  the  area  of  the  segment? 

6.  Find  the  area  of  the  segment  if  the  area  of  sector  AOB  is 
98.3  and  triangle  AOB  76.84. 

6.  State  how  we  can  find  area  of  a  segment  of  a  circle. 

7.  Find  the  area  of  each  of  the  segments  in  the  following, 
having  given  that  the  angle  AOB  is  60°: 

Radius  of  circle  8;  9;  10;  12;  15;  22;  30;  42. 


119 


LESSON  LXII 
REVIEW 


1.  The  radius  of  each  circle  is  5. 
between  the  circles  of  Fig.  66. 


Find  the  area  included 


FIG.  66. 


FIG.  67. 


2.  The  diameter  of  a  circle  is  10.     The  circumference  is 
divided  into  six  equal  parts  and  lobes  formed  as  in  figure.    Find 
the  area  of  each  lobe  of  Fig.  67. 

3.  The  diameter  AE  of  Fig.  68  is  divided  into  four  equal 
parts,  and  semicircles  drawn  as  indicated.     Find  the  area  of 
each  figure.    Let  AE  equal  8. 


FIG.  68. 


FIG.  69. 


4.  Find  the  area  of  each  lobe  of  Fig.  69  if  the  side  of  the 
square  is  16. 


LESSON  LXIII 
REVIEW 

1.  One  cubic  inch  of  steel  weighs  .29  Ib.  An  I-beam  has  a 
cross-section  as  shown  in  Fig.  70  and  a  length  of  12  in.  Find 
its  weight. 


-72- 


FIG.  70. 

2.  In  forging  a  bolt  24  in.  in  total  length  i^  in.  in  diameter 
with  a  head  ^  in.  thick  and  i^  in.  square,  the  stock  is  cut 
from  a  bar  of  iron,  i^  in.  square  in  cross-section.     How  long  a 
piece  will  it  take?     Allow  i  in.  in  length  for  waste. 

3.  A  circular  disk  ^  in.  thick  and  4  in.  in  diameter  is  to  be 
made  from  the  same  rod.     Find  the  length  required. 

4.  The  external  diameter  of  a  hollow  cast-iron  shaft  is  18  in. 
and  its  internal  diameter  is  10  in.     Calculate  its  weight  if  the 
length  is  20  ft.  and  cast  iron  weighs  .26  Ib.  per  cu.  in. 

6.  Find  the  length  of  steel  wire  in  a  coil,  if  its  diameter  is 
.025  in.,  and  its  weight  50  Ib. 

6.  The  larger  diameter  of  a  piece  of  steel  is  %  in.,  and  the 
smaller  3^4  in.     Find  the  taper  per  foot  if  the  piece  is  i%  ft. 
long. 

7.  The  larger  diameter  of  a  piece  of  steel  is  3^  in.     If  it  is 
i%  ft.  long  and  the  taper  is  %  in.  per  ft.,  what  is  the  smaller 
diameter?     Find  the  offset  for  turning  this  taper;  also  the 
taper  of  problem  6. 

121 


122  INDUSTRIAL   ARITHMETIC 

8.  Find  the  cost  of  25  pieces  of  2  in.  X  8  in.  each  16  ft. 
long  at  $30  per  M. 

9.  It  is  required  to  build  a  bin  that  will  hold  50  bu.     It  can 
be  built  in  a  space  7  ft.  long  and  3  ft.  6  in.  wide.     How  high 
must  it  be  if  a  bushel  contains  \Y±  cu.  ft.? 

10.  A  certain  lumber  company  has  a  piece  of  timber  3  ft. 
square  and  80  ft.  long.     How  many  board  feet  in  the  piece? 


LESSON  LXIV 
REVIEW 

1.  Find  the  area  between  the  tangents  to  the  circle  and  the 
arc.  The  tangents  make  an  angle  of  60°,  the  radii  are  per- 
pendicular to  the  tangents  and  OF  bisects  angle  APB.  OA  = 
12  in. 


2.  Find  the  area  included  between  the  four  circles, 
radius  is  5.     ABCD  is  a  square. 


Each 


FIG.  72. 


FIG.  73- 


3.  The  diameter  of  the  circle  is  16.     Semicircles  are  drawn  as 
indicated  in  Fig.  73.     Find  area  of  shaded  portion. 

4.  The  center  of  one  circle  lies  on  the  circumference  of  the 

123 


I24 


INDUSTRIAL   ARITHMETIC 


other.     If  the  radius  of  each  circle  is  12,  find  the  area  common 
to  both  circles.     Fig.  74. 


FIG.  75. 


5.  Find  the  area  of  the  two  crescents  formed  as  given  in  Fig. 
75.     ACB  is  a  right  triangle. 


LESSON  LXV 
AREA  OF  THE  SURFACE  OF  A  PYRAMID  AND  OF  A  CONE 

The  line  from  the  vertex  of  a  pyramid  perpendicular  to  its 
base  is  called  the  altitude  of  the  pyramid.  The  foot  of  this 
altitude  in  such  pyramids  as  we  shall  study  is  the  middle  of 
the  base.  Name  the  altitude  of  the  pyramid  (Fig.  76). 


FIG.  77. 


The  altitude  of  any  one  of  the  triangles  that  form  the  faces 
of  the  pyramid  is  called  the  slant  height  of  the  pyramid. 
What  is  the  slant  height  of  the  above  pyramid? 

What  is  the  altitude  of  the  cone?     The  slant  height? 

The  area  of  the  curved  surface  of  a  cone  is  one-half  the  prod- 
uct of  its  slant  height  by  the  circumference  of  its  base?  How 
do  you  find  the  area  of  the  surface  of  a  pyramid? 

Find  the  area  of  the  lateral  surface  of  each  of  the  following 
pyramids: 

1.  Base  a  square  20  in.  each  side,  slant  height  18  in. 

2.  Base  a  square  18  in.  each  side,  slant  height  20  in. 

"5 


126  INDUSTRIAL   ARITHMETIC 

3.  Base  a  square  4  ft.  5  in.  each  side,  slant  height  3  ft.  8  in. 

4.  Base  an  equilateral  triangle  each  side  a,  slant  height  b. 

5.  Find  the  complete  area  of  each  of  the  above  pyramids. 

6.  If  each  side  of  the  base  of  a  square  pyramid  is  8  in.  and  its 
altitude  6  in.,  what  is  its  slant  height?     Its  area? 

7.  Find  the  area  of  the  curved  surface  of  each  of  the  follow- 
ing cones: 

Radius  of  base  12  in., slant  height  16  in.;  radiusof  base  10  in., 
slant  height  20  in.;  radius  of  base  3  ft.,  slant  height  10  ft.; 
radius  of  base  2  ft.,  slant  height  4  ft.;  radius  of  base  5  ft.  4  in., 
slant  height  12  ft.  6  in.;  radius  of  base  4  ft.  5  in.,  slant  height 
8  ft.  9  in. 

8.  The  altitude  of  a  cone  is  8  in.  and  the  radius  of  its  base 
6  in.     What  is  its  slant  height?     Its  area? 

9.  The  slant  height  of  a  cone  is  i  ft.  8  in.,  and  the  radius  of 
its  base  6  in.     What  is  the  area  of  its  surface?    Its  complete 
area?     Its  altitude? 

10.  Find  the  complete  area  of  a  cone  whose  slant  height  is 
24  in.,  the  radius  of  its  base  being  8  in. 

11.  The  radius  of  the  base  of  a  cone  is  5  in.,  and  its  slant 
height  makes  an  angle  of  60°  with  the  radius.     Find  the  com- 
plete area  of  the  cone. 


LESSON  LXVI 
VOLUME  OF  A  PYRAMID  AND  OF  A  CONE 

A  pyramid  is  one-third  of  a  prism  with  the  same  base  and 
altitude  as  the  prism. 

A  cone  is  one-third  of  a  cylinder  with  the  same  base  and  alti- 
tude as  the  cylinder. 

How  then  will  you  find  the  volume  of  a  pyramid?  Of 
a  cone? 

1.  Find  the  volume  of  each  of  the  following  pyramids: 
Base  a  square  10  ft.  on  each  side,  altitude  15  ft. ;  base  a  square 

123^5  ft.  each  side,  altitude  24  ft.;  base  a  square  9  ft.  8  in.  each 
side,  altitude  10  ft.;  base  a  square  7  ft.  3  in.  each  side,  altitude 
7  ft.,  base  an  equilateral  triangle  each  side  10  in.  and  altitude 
20  in. 

2.  What  is  the  volume  of  a  pyramid  whose  altitude  is  18  ft. 
6  in.  and  whose  base  is  a  right  triangle,  hypothenuse  10  ft.  and 
one  acute  angle  30°? 

3.  What  is  the  weight  of  a  solid  steel  pyramid  18  in.  long 
and  the  base  a  square  10  in.  on  each  side? 

4.  Find  the  volume  of  each  of  the  following  cones: 
Radius  of  base  8  in.,  altitude  12  in.;  radius  of  base  12  in., 

altitude  15  in.;  radius  of  base  10  in.,  altitude  3  ft.;  radius  of 
base  3  ft.  3  in.,  altitude  i  ft.  6  in.;  radius  of  base  3  ft.,  altitude 
3  ft.;  radius  of  base  4  in.,  altitude  4  ft.  2  in.;  radius  of  base  r, 
altitude  h. 

6.  Find  the  volume  in  cubic  inches  of  the  following  round 
piece : 

127 


128  INDUSTRIAL   ARITHMETIC 


FIG.  78. 

6.  A  cylindrical  piece  of  steel  i  ft.  4  in.  long  and  4  in.  in 
diameter  has  a  conical  hole  4  in.  long  and  3  in.  in  diameter 
bored  from  one  end  of  it.     What  is  the  weight  of  the  piece? 

7.  Find  the  volume  of  a  cone  10  in.  in  diameter  if  it  tapers 
^  in.  per  in. 

8.  Find  the  volume  of  a  cone  i  ft.  6  \n.  long  if  it  tapers  i  in. 
per  in. 


LESSON  LXVII 
REVIEW 

1.  A  pile  of  coal  of  conical  shape  10  ft.  high  lies  at  an  angle 
of  30°  with  the  horizontal.     How  many  tons  in  it  if  i  cu.  ft. 
weighs  38  lb.? 

2.  Find  the  weight  of  a  conical  casting  of  iron  8  in.  in  diame- 
ter and  slant  height  14  in. 

3.  The  rain  which  falls  on  a  house  22  ft.  X  36  ft.  is  con- 
ducted to  a  cylindrical  cistern  8  ft.  in  diameter.     How  great  a 
fall  of  rain  would  it  take  to  fill  the  cistern  to  a  depth  of  7^  ft.  ? 

4.  How  many  gallons  of  water  will  a  6-in.  pipe  deliver  per 
hour  if  the  flow  is  3  ft.  per  sec.? 

5.  A  band  saw  runs  on  pulleys  48  in.  in  diameter  at  a  rate  of 
1 80  R.P.M.     If  the  pulleys  are  decreased  18  in.  in  diameter, 
how  many  R.P.M.  will  they  have  to  make  to  keep  the  band  saw 
travelling  at  the  original  speed?  . 

6.  A  shaft  has  upon  it  two  pulleys,  each  8  in.  in  diameter. 
The  speed  of  the  shaft  is  400  R.P.M.     What  must  be  the 
size  of  the  pulleys  of  two  machines  if,  when  belted  to  these 
shaft  pulleys,  one  of  them  has  a  speed  of  300  R.P.M.  and  the 
other  900? 

7.  A  %-in.  drill,  cutting  cast  iron,  may  cut  at  the  rate  of  40 
ft.  per  min.     How  many  R.P.M.  may  it  make? 

8.  Find  the  cost  at  40$.  per  lb.  for  sheet  copper  to  line 
bottom  and  sides  of  a  cubical  vessel  7  ft.  each  edge,  if  the  sheet 
copper  weighs  12  oz.  per  sq.  ft. 

9.  If  the  feed  is  % Q  in.  and  the  work  has  a  speed  of  164  ft. 
per  min.,  how  long  will  it  take  to  cut  a  piece  2  in.  in  diameter 
and  i  ft.  long? 

9  129  , 


130  INDUSTRIAL   ARITHMETIC 

10.  If  S  =  S'  and  L  =  %-in.  pitch,  find  the  simple  gearing 
that  may  be  used  to  cut  15^  threads  per  in. 

11.  If  the  threads  of  problem  10  is  a  60°  V  thread,  find  its 
depth. 

12.  The  ratio  of  the  areas  of  two  circles  is  i :  4  and  the  radius 
of  the  smaller  circle  is  6.     What  is  the  radius  of  the  larger 
circle? 

13.  Studding  for  partitions  is  2  in.  X  4  in.  and  16  ft.  long. 
It  is  set  1 6  in.  between  centers.     How  many  pieces  must  be 
bought  for  a  partition  8  ft.  high  and  12  ft.  long?     What  will  it 
cost  at  $30  per  M  ? 

14.  A  right  cone  of  altitude  10  ft.  has  a  slant  height  of  18  ft. 
Find  its  complete  area  and  also  its  volume.   . 


LESSON  LXVIII 
REVIEW 

1.  A  main  line  shaft  runs  176  R.P.M.,  a  pulley  on  this  shaft 
is  36  in.  in  diameter  and  is  belted  to  a  pulley  on  the  counter- 
shaft 12  in.  in  diameter.     Another  pulley  on  this  same  counter- 
shaft is  16  in.  in  diameter  and  is  belted  to  a  pulley  4  in.  in 
diameter  on  a  grinder.     What  is  the  speed  of  the  counter- 
shaft?    Of  grinder  spindle?     If  the  grinding  wheel  is  10  in.  in 
diameter,  what  is  its  surface  speed? 

2.  The  diameter  of  a  driving  pulley  is  9  in.  and  its  speed  is 
1000  R.P.M.     What  is  speed  of  driven  pulley  whose  diameter 
is  4  in.  ?     If  this  speed  is  too  fast,  what  should  be  the  diameter 
of  the  diven  pulley  to  have  a  speed  250  R.P.M.  less  than  the 
4-in.  pulley?     If  the  speed  of  the  4-in.  pulley  is  too  slow  by  250 
R.P.M.,  what  size  driving  pulley  should  be  used  instead  of  the 
9-in.  pulley?     If  we  keep  both  pulleys  (9  in.  and  4  in.)  and  make 
our  speed  changes  by  changing  speed  of  9-in.  pulley,  what 
would  be  the  speed  of  9-in.  pulley  to  give  1125  R.P.M.  of  4-in. 
pulley? 

3.  How  many  gallons  of  water  in  a  railway  track  tank  1200 
ft.  long,  19  in.  wide  and  7  in.  deep,  if  the  water  is  2  in.  below  the 
top  of  the  tank? 

4.  The  diameters  of  the  steps  of  a  step  cone  pulley  are  8  in., 
5^  in.  and  4  in.  respectively.     Find  the  ratio  of  their  surface 
speeds  when  the  shaft  to  which  the  pulley  is  attached  is  making 
900  R.P.M. 

5.  A  pump  has  a  water  cylinder  of  6  in.  and  a  stroke  of  16  in. 
How  many  gallons  of  water  are  pumped  in  i  hr.  if  -the  pump 
makes  60  strokes  per  min.? 


132  INDUSTRIAL   ARITHMETIC 

6.  A  smokestack  90  ft.  high  is  to  be  held  in  place  by  five 
guy  wires  attached  30  ft.  from  the  top  of  the  stack.     The  wires 
are  anchored  55  ft.  from  the  bottom  of  the  stack  on  a  level  with 
its  bottom.     Find  the  number  of  feet  in  the  guy  wires  allowing 
35  ft.  for  fastening. 

7.  One  formula  for  making  concrete  is  i  part  cement,  2 
parts  sand  and  4  parts  crushed  stone.     How  many  cubic  feet 
of  each  will  be  required  to  make  a  concrete  wall  100  ft.  long,  2 
ft.,  thick  and  4  ft.  high. 

8.  What  is  the  micrometer  reading  for  a  piece  of  iron  whose 
diameter  is  .422?     About  how  many  64ths  of  an  inch? 

9.  Each  edge  of  a  pyramid  of  four  faces  is  8  in.     Find  its 
complete  area. 

10.  Find  the  value  of  the  letter  in  each  of  the  following: 

• 

3X  +  8x  -  6x  +  4X  =  25;  >^x  +  >£x  +  %x  =  10; 
yx  +  2X  +  3x  -  8x  =  15;  MX  +  >£t  +  Kox  =  30; 
2X  +  3X  +  8x  —  2X  =  14;  Y&  +  %x  +  x  =28. 


LESSON  LXIX 
REVIEW 

1.  A  steel  plate  5  ft.  long,  3  ft.  6  in.  wide  and  i  in.  thick,  has 
a  hole  10  in.  in  diameter  cut  through  it.     Find  weight  of  plate, 
allowing  .29  Ib.  per  cu.  in. 

2.  A  tubular  boiler  has  124  tubes  each  3%  in.  in  diameter 
and  1 8  ft.  long.     What  is  the  total  tube  surface? 

3.  A  room  is  heated  by  steam  pipes.     There  are  240  ft.  of 
2-in.  pipes  and  52  ft.  of  5-in.  pipes  and  2  ft.  of  4^-in.  feed  pipe. 
What  is  the  total  heating  surface  of  the  room? 

4.  A  hollow  steel  shaft  10  ft.  long  is  18  in.  in  external  diame- 
ter and  8  in.  in  internal  diameter.     Find  weight  of  the  shaft. 

5.  If  the  depth  of  a  sharp  V  thread  angle  60°  is  %  in.,  what  is 
its  pitch? 

•  6.  Three  circles  each  of  g-in.  radius  are  tangent  to  each  other. 
Find  area  between  the  three  circles. 

7.  Find  the  volume  generated  by  an  equilateral  triangle 
whose  side  is  8  if  it  revolves  about  its  altitude  as  an  axis.     Find 
area  also. 

8.  If  the  back  stay  of  a  suspension  bridge  is  125  ft.  long  and 
is  anchored  1 20  ft.  from  the  base  of  the  pier,  what  is  the  height 
of  the  pier? 

9.  The  distance  across  the  flats  of  a  2-in.  Whitworth  hexagon 
nut  is  3^2  m-     Find  the  distance  across  the  corners. 

10.  A  pipe  has  a  sectional  area  of  125  sq.  in.  at  i  part  and 
80  sq.  in.  at  another.    If  6000  cu.  ft.  of  water  flow  past  each 
section  per  hour,  find  the  velocity  of  the  water  in  feet  per 
second,  at  each  point. 

11.  A  triangular  piece  of  steel  is  %  in.  thick  and  two  of  its 
sides  are  28 -in.  and  32  in.     If  these  sides  form  an  angle  of  30°, 
what  is  the  weight  of  the  piece? 

12.  Solve  each  of  the  following: 

6a2  +  10  =  160;  2b2  +  3b2  =  500;  8b2  —  100  =  700; 
ion2  —  10  =  90;     8k2  —  15    =  225;  6x2  +  4      =40. 

133 


LESSON  LXX 
REVIEW 

1.  Find  the  weight  of  100  steel  planer  bolts  with  heads  i  in. 
square  and  %  in.  thick,  body  5  in.  long  and  diameter  %  in. 
2. 


FIG.  79. 

Calculate  the  amount  of  stock  in  the  angle  weld  (Fig.  79). 
3. 


FIG.  80. 

Calculate  the  length  of  stock  J^  in.  X  i  in.  required  for 
the  forging  shown  in  the  diagram. 


REVIEW 


135 


4. 


FIG.  81. 


Find  the  weight  of  the  forging  and  the  length  of  stock  2  in. 
X  4  in.  required  to  make  it.     i  cu.  in.  weighs  .2779  Ib. 
5. 


.._ 

T 

} 

xl" 

J 

<                        n"                    * 

„ 

'    'i    1 

'  7  ' 

*  2 

' 

"*         4         * 

"*                                                 70                                                    *• 

FIG.  82. 


Find  the  weight  of  the  forged  crankshaft  as  per  diagram. 
6. 


FIG.  83. 
Calculate  the  amount  of  stock  in  the  eye  bend. 


136 
7. 


INDUSTRIAL   ARITHMETIC 


-*%- 


FIG.  84. 

The   link  is  forged   from  %6-in.  round  stock.     Find  the 
weight  of  a  chain  of  50  steel  links. 


LESSON  LXXI 
REVIEW 

1.  An  engine  pumping  water  from  a  cylindrical  tank  10  ft. 
6  in.  in  diameter  lowered  the  surface  of  the  water  2  ft.  4  in. 
What  was  the  weight  of  the  water  pumped  out? 

2.  What  is  the  weight  of  a  piece  of  steel  shafting  12  ft.  long 
and  3  in.  in  diameter? 

3.  A  brass  plate  2  in.  thick,  in  the  shape  of  a  semicircle  with 
a  radius  of  14.5  in.,  has  four  holes  through  it,  each  %  m-  in 
diameter.     What  is  its  weight?     i  cu.  in.  of  brass  weighs 
.3031  Ib. 

4.  A  rectangular  box  is  8  ft.  8  in.  long,  5  ft.  2  in.  wide  and  4 
ft.  3  in.  high.     Find  the  cost  at  i2)£.  per  Ib.  of  lining  the  sides 
and  bottom  with  lead  weighing  7  Ib.  per  sq.  ft. 

5.  Two  pulleys  each  18  in.  in  diameter  are  connected  with  a 
belt.     How  long  is  the  belt  if  the  distance  between  their  centers 
is  10  ft.? 

6.  The  water  cylinder  of  a  pump  is  6^4  m-  m  diameter  and 
the  length  of  the  stroke  is  15^  in.     Find  the  time  such  a  pump 
would  require  to  empty  a  rectangular  cistern  30  ft.  6  in.  long, 
13  ft.  9  in.  wide  and  12  ft.  4  in.  deep,  if  the  pump  is  making  90 
strokes  per  min.? 

7.  The  angle  of  elevation  of  the  top  of  a  flag  staff  is  30°  at  a 
point  P.     If  the  distance  from  P  to  the  foot  of  the  staff  is  60  ft., 
how  high  is  the  staff?     • 

8.  When  a  house  is  heated  with  a  hot  air  furnace  the  cross- 
sectional  area  of  the  cold  air  box  should  equal  three-fourths  the 
total  cross-sectional  area  of  the  hot  air  pipes  supplying  the 
rooms?     What  should  be  the  area  of  the  cold  air  box  for  eight 
hot  air  pipes,  two  being  10  in.  in  diameter  and  the  others  each 
8  in.  in  diameter? 

137 


LESSON  LXXII 
REVIEW 

1.  A  poker  handle  is  made  from  ^-in.  stock.     The  ring  is 
i  in.  in  diameter  inside  measurement.     How  much  stock  must 
be  allowed  for  the  making  of  ring?     The  length  is  calculated 
from  center  of  stock. 

2.  The  wheels  of  a  band  saw  are  36  in.  in  diameter.     What  is 
the  speed  in  feet  per  minute  if  the  wheels  are  making  500 
R.P.M.;  if  3ooR.P.M.? 

3.  If  the  small  diameter  of  a  taper  shank  is  %  in.  and  the 
taper  is  .6  in.  per  ft.,  what  would  be  the  large  diameter  of  the 
shank  if  it  were  4  in.  long? 

4.  If  a  forge  uses  30  Ib.  of  coal  per  day  and  there  are  23  forges 
in  the  shop,  what  is  the  cost  of  fuel  at  $4.50  per  ton  to  run  the 
shop  200  school  days? 

6.  How  many  feet  per  minute  does  a  point  on  the  surface  of 
a  2^-in.  cylinder  travel,  if  the  cylinder  is  making  1200  R.P.M.? 
600?  3000? 

Find  the  cost  of  the  following  order  for  lumber  used  in  mak- 
ing imposing  tables. 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

16 

Chestnut 

3^"X  3^"X37" 

it- 

8 

Chestnut 

%"XioM"X34" 

/ 

it- 

8 

Chestnut 

%"XioM"X22" 

it- 

8 

Chestnut 

%"X3^"  X34" 

it- 

8 

Chestnut 

%"X3M"  X22" 

7*. 

4 

Chestnut 

%"X24"     X36" 

It. 

8 

Chestnut 

%"X6M"  X24" 

it- 

8 

Poplar 

%"X6M"  Xa4" 

6i. 

4 

Poplar 

%"X6W'  X24" 

6i. 

4 

Poplar 

M"X24"      X24" 

64. 

138 


REVIEW 


139 


Find  the  total  number  of  board  feet  and  the  cost  of  the  fol- 
lowing mill  bill  for  doors  for  a  case  in  the  stock  room. 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

H 

Chestnut 

i"X2^"X66^" 

it- 

8 

Chestnut 

i"X2K"Xso" 

it. 

8 

Chestnut 

i"X2^"X37" 

it- 

22 

Chestnut 

i"X2K"Xi4K" 

it- 

IS 

Chestnut 

i"Xs"     Xi4K" 

It- 

Find  the  total  number  of  board  feet  in  the  following  mill  bill 
for  indian  clubs: 


No.  of  pieces 

Wood 

Dimensions 

Bd.  ft. 

Price 

Cost 

2OO 

400 

Chestnut 
Chestnut 

2"     X4M"Xi9" 
iK"X3%"X8" 

It- 
it- 

LESSON  LXXIII 
ALLOYS 

An  alloy  is  a  mixture  obtained  by  fusing  metals  with  each 
other.  When  two  or  more  metals  are  fused  we  obtain  a  new 
metal  that  often  has  properties  exhibited  by  none  of  the  metals 
in  the  combination  nor  by  any  single  metal. 

For  example,  gold,  which  in  its  pure  state  is  too  soft  and 
flexible  for  use  as  coins,  jewelry,  etc.,  is  hardened  by  mixing 
with  copper  in  the  ratio  of  9  parts  gold  to  i  part  of  copper. 
Brass  is  an  alloy  of  copper  and  zinc,  harder  than  copper  and 
easier  to  work. 

Unless  a  small  quantity  of  lead  is  added  to  brass  it  cannot  be 
used  in  turning  operations,  since  the  tool  will  tear  and  not 
cut  it.  An  alloy  of  50%  bismuth,  30%  lead  and  20%  tin  has 
the  property  of  melting  at  a  much  lower  temperature  than  any 
of  the  metals  in  the  combination. 

In  industrial  work  metals  that  have  certain  peculiar  proper- 
ties, are  needed  for  special  purposes.  Type  metal  must  have 
the  property  of  making  sharp  distinct  lines;  likewise  pattern 
metal.  This  is  accomplished  by  adding  antimony  to  the  alloy. 
Antimony  when  used  as  a  part  of  the  alloy  causes  the  metal  to 
expand  on  cooling;  hence  to  fill  out  the  corners  of  the  molds 
making  a  pattern  with  sharp  lines. 

Antimony  also  renders  the  alloy  harder.  The  bearings  for 
machinery  must  be  antifrictional,  that  is,  not  easily  heated 
by  contact  with  the  revolving  machinery.  They  must  be  hard 
and  strong  particularly  for  heavy  work;  for  high-speed  machin- 
ery softer  bearings  may  be  used. 

A  fairly  good  bearing  for  high-speed  machinery  contains 

140 


ALLOYS  .  141 

80%  lead  and  20%  tin,  but  is  not  very  hard.  A  metal  widely 
used  for  heavy  machinery  bearings  contains  80.5%  lead, 
11.5%  tin,  7.5%  antimony  and  .5%  copper.  If  a  metal  is 
required  that  melts  at  a  low  temperature  bismuth  is  used  as 
part  of  the  alloy.  The  plugs  of  automatic  sprinklers  and  doors 
used  for  fire  protection  are  made  of  such  an  alloy. 

Alloys  containing  sodium  have  the  peculiar  property  of 
producing  by  oxidation,  material  which  will  saponify  with  the 
oil  used  in  the  bearing  and  thus  assist  lubrication.  Thus  by 
combining  copper,  lead,  zinc,  etc.,  in  different  ratios  we  often 
produce  a  new  metal  that  answers  the  peculiar  needs  of  various 
industrial  problems. 

Exercises 

1.  In  the  foundry  the  formula  for  "yellow  brass  castings" 
contains  66%%  copper  and  33^%  zinc.     How  many  pounds 
of  each  must  be  used  to  make  1500  Ib.  of  yellow  brass? 

2.  How  many  pounds  of  copper  are  used  with  200  Ib.  of 
zinc  to  make  yellow  brass? 

3.  Recently  in  your  foundry  44  Ib.  of  copper,  3  Ib.  zinc,  2  Ib. 
tin  and  i  Ib.  of  lead  were  used  to  make  a  bronze  casting.     De- 
termine the  per  cent,  of  each  element  used. 

4.  How  many  pounds  of  each  element  must  be  used  to  make 
1800  Ib.  of  bronze  according  to  the  formula  in  problem  3? 

5.  How  many  pounds  of  bronze  will  100  Ib.  of  tin  make? 

6.  In  making  pattern  metal  we  use  50%  tin,  45%  zinc  and 
5%  antimony.     How  much  pattern  metal  will  90  Ib.  of  zinc 
make?    28  Ib.  of  antimony?    500  Ib.  of  tin? 

7.  A  pattern  that  weighs  10  Ib.  will  contain  how  much  of 
each  of  the  elements  of  pattern  metal? 

'8.  Fire  sand  is  about  98%  silica.     What  is  the  weight  of  the 
silica  in  5000  Ib.  of  fire  sand? 

9.  A  pile  of  fire  sand  contains  i  ton  of  silica.  How  many 
tons  of  sand  in  the  pile? 


142  INDUSTRIAL    ARITHMETIC 

10.  A  certain  grade  of  molder's  sand  is  76%  silica  and  4% 
aluminum.     How  much  of  the  sand  will  contain  758  Ib.  of 
silica?    How  much  aluminum  in  the  sand? 

11.  Another  grade  of  molder's  sand  is  86%  silica  and  8% 
aluminum.     How  much  silica  in  a  pile  of  the  sand  that  contains 
872  Ib.  of  aluminum? 

12.  One  of  the  bronze  castings  made  in  your  foundry  weighed 
28.5  Ib.     How  many  pounds  of  tin  in  it?     Of  lead? 

13.  A  certain  alloy  contains  50  Ib.  of  copper  and  25  Ib.  of 
zinc.     Write  the  formula  for  this  alloy  in  per  cents. 


LESSON  LXX1V 
ALLOYS 

1.  How  many  pounds  of  carbon  in  a  ton  of  cast  iron  that 
contains  2.75%  carbon? 

2.  If  tool  steel  is  1.25%  carbon  how  many  pounds  of  tool 
steel  will  contain  25  Ib.  of  carbon? 

3.  Naval  brass  is  62%  copper,  i%  tin  and  37%  zinc.     Find 
the  amount  of  each  in  1200  Ib.  of  naval  brass.     U.  S.  Navy 
Department. 

4.  Hard  bronze  for  piston  rings  is  22%  tin  and  78%  cop- 
per.    How  many  pounds  of  hard  bronze  will  contain  2340  Ib. 
of  carbon?     U.  S.  Navy  Department. 

6.  An  alloy  of  88%  tin,  8%  antimony,  3.5%  copper  and  .5% 
bismuth  is  used  for  the  bearings  of  high-speed  dynamos.  Cal- 
culate the  amount  of  each  in  2450  Ib.  of  such  alloy. 

6.  The  bearings  for  railway  trucks  contain  42%  tin,  56% 
zinc  and  2%  copper.     How  much  tin  and  copper  must  be  used 
to  make  such  an  alloy  that  contains  672  Ib.  of  zinc? 

7.  Babbitt  metal  for  high  pressure  bearings  is  90%  tin, 
7%  antimony  and  3%  copper.     Find  the  amount  of  antimony 
and  copper  in  such  a  composition  that  contains  1800  Ib.  of  tin. 

8.  One  ton  of  babbitt  metal  adapted  for  low  pressure  and 
medium  speed  contains  160  Ib.  tin,  400  Ib.  antimony  and  1440 
Ib.  lead.     Find  the  per  cent,  of  each  for  this  composition. 

9.  One  thousand  pounds  of  plastic  metal  contain  800  Ib.  tin, 
100  Ib.  lead,  10  Ib.  antimony,  80  Ib.  copper,  and  10  Ib.  bismuth. 
Find  the  per  cent,  of  each  in  this  alloy. 

10.  Find  the  amount  of  each  element  in  2500  Ib.  of  the 
alloys  in  problems  8  and  9. 

11.  The  U.  S.  Navy  Department  uses  brazing  metal  that 

143 


I 

144  .  INDUSTRIAL   ARITHMETIC 

is  85%  copper  and  15%  zinc.     How  many  pounds  of  each  will 
be  required  to  make  3200  Ib.  of  this  alloy? 

12.  At  $13  per  100  Ib.  for  copper  and  $5  for  zinc,  find  the 
cost  of  the  alloy  in  problem  n. 

13.  Use  the  following  to  find  the  cost  of  as  many  different 
alloys  given  above  as  possible:  Cost  per  100  Ib. — lead  $4; 
zinc  $5;  antimony  $9;  copper  $13;  and  tin  $30. 


LESSON  LXXV 


THE  PRINT  SHOP 

The  units  of  length  in  the  print  shop  are  the  inch,  the  pica, 
the  nonpareil  and  the  point. 

The  dimensions  of  cards,  sheets  of  paper,  etc.,  are  expressed  in 
inches.  The  pica,  which  is  ^  in.  is  used  to  measure  the  lengths 
of  printed  matter,  e.g.,  the  dimensions  of  a  piece  of  printed 
matter  is  usually  expressed  as  13  picas  wide  and  20  picas  long, 
or  20  picas  wide  by  30  picas  long,  etc.  Sometimes,  however, 
the  width  is  expressed  in  picas  and  the  length  in  inches,  e.g., 
a  newspaper  column  13  picas  wide  and  20  inches  long. 

The  nonpareil  is  one-half  of  a  pica,  and  the  point  is  one- 
twelfth  of  a  pica. 


flonp. 


FIG.  85. 

The  body  of  metal  type  is  measured  by  the  point,  as  i6-point 
type,  36-point  type,  etc.  A  24-point  type  means  the  height 
of  the  type  is  24  points  or  2^f2  m-  Metal  type  is  made  in  the 
following  number  of  points: 

Common  sizes:  6,  8,  10,  12,  14,  18,  24,  30,  36,  48,  60,  72,  84, 
96,  1 20. 

Odd  sizes,  chiefly  book  and  newspaper  sizes:  3^3,  4,  4^,  5, 
5>£,  7,  9,  ii. 

Sizes  rarely  used:  16,  20,  22. 

e-poiNT  10-POINT  12-POINT 

FIG.  86. 

10  145 


146  INDUSTRIAL   ARITHMETIC 

The  amount  of  type  in  any  composition  is  measured  by  a 
square  whose  side  is  any  number  of  points.  This  unit  is  called 
the  EM.  The  number  of  ems  in  any  body  of  printed  matter 
corresponds  to  the  area  of  a  rectangle.  When  measuring 
printed  matter  set  in  8-point  type  the  side  of  the  em  is  8 
points,  set  in  lo-point  type  the  side  of  the  em  is  10  points,  etc. 

Exercises 

1.  How  many  picas  in  3  in.?     4)-^  in.?     %  in.?     6  in.? 
24  in.?     How  many  nonpareils?    How  many  points? 

2.  In  the  following  number  of  points  find  the  number  of 
picas,  inches,  and  nonpareils: 

360;  24;  320;  144;  100;  168. 

3.  Find  the  number  of  square  points  in  an  8-point  em;  a 
lo-point  em,  a  6-point  em;  an  a-point  em;  a  y-point  em. 

4.  What  is  the  value  of  each  of  the  following  ratios: 
8-point  em:  6-point  em;  lo-point  em:  1 6-point  em;  12 -point 

em:  3 6-point  em? 

5.  How  many  8-point  ems  in  each  of  the  following  ems: 
640  lo-point  ems?    3200  i2-point  ems?    1280  1 6-point  ems? 

6.  At  the  same  rate  per  M  which  will  cost  the  most,  a  page 
set  in  8-point  type  or  lo-point  type  or  i6-point  type?     Why? 

7.  An  edition  of  a  certain  newspaper  had  seven  columns  13 
picas  wide  and  21  in.  long  on  each  page.     How  many  i2-point 
ems  per  page?     How  many  lo-point  ems? 

8.  The  body  of  the  printed  matter  on  the  page  of  a  certain 
book  is  20  picas  wide  and  33  picas  long.     How  many  4-point 
ems  per  page?     Find  the  cost  of  setting  the  page  at  50^.  per 
M  8-point  ems. 

9.  A  double  newspaper  column  is  26^  picas  wide  and  24  in. 
long.     Find  the  cost  of  setting  2)^  such  columns  at  54^.  per 
M  8-point  ems. 

10.  How  many  ems,  and  what  size  on  the  page  of  this  book? 
Answer  the  same  questions  about  your  text-book  in  English. 


PRINT   SHOP  147 

11.  If  130  ems  of  composition  contain  50  words,  how  many 
ems  in  an  article  of  2000  words?     2500  words?     1600  words? 

12.  How  many  lines  set  solid  of  i8-point  type  in  i  in.?     In  4 
in.?     In  12  in.?     In  5^  in.?     In  10%  in.?     In  a  in.?     In 
b  in.?     In  an  ordinary  newspaper  column? 

13.  How  many  lines  per  inch  set  solid  of  the  sizes  of  type  in 
common  use? 

14.  What  size  type  is  used  to  set  solid  9  lines  per  in.?     7 
lines  per  in.?     12  lines  per  in.?     2  lines  per  in.? 

15.  Find  the  width  of  a  printed  sheet  of  seven  columns  12^ 
picas  wide  with  a  margin  of  %  in.  on  each  side  if  the  columns 
are  spaced  with  6-point  rule. 

16.  Measure  several  of  your  text-books  to  determine  the 
size  of  the  type  used. 


LESSON  LXXVI 
PRINT  SHOP 

1.  A  sheet  of  cardboard  22^  in.  X  28^2  in.  is  to  be  cut  into 
tickets  2^  in.  X  4  in.     Find  the  greatest  number  of  tickets 
that  can  be  cut  from  the  piece  and  tell  how  you  will  cut  it. 

2.  How  many  sheets  of  cardboard  22^  in.  X  28^  in.  will 
be  required  to  cut  1500  tickets  2^  in.  X  4  in.?     If  the  stock 
costs  $2.50  per  100  sheets,  how  much  will  the  tickets  cost? 

3.  Your  schedule  cards  are  4  in.  X  5%  in.  and  were  cut  from 
cardboard   22^   in.    X    28^   in.     How  many  sheets  were 
required  to  make  5000  of  these  cards?     Find  their  cost  at 
$2.40  per  100  sheets. 

4.  How  many  cards  2%  in.  X  4%  in.  can  be  cut  from  50 
sheets  of  the  cardboard  used  in  problem  3  ? 

6.  A  card  2  in.  X  7  in.  was  cut  from  cardboard  22  in.  X  28  in. 
If  the  cardboard  costs  $2.40  per  100  sheets,  find  the  cost  of  the 
cards  per  1000. 

6.  Answer  each  of  the  above  questions  allowing  10%  for 
press  waste  for  each. 

7.  Letter  paper  sheets  16  in.  X  21  in.  are  cut  into  letter 
heads  8  in.  X  10^  in.     How  many  sheets  will  be  required  for 
2M  such  letter  heads  allowing  10%  for  press  waste?    What 
will  the  paper  cost  at  $2.25  per  ream  of  500  sheets? 

8.  Which  cuts  to  the  better  advantage  sheets  17  in.  X  22  in. 
or  sheets  16  in.  X  21  in.,  if  letter  heads  8  in.  X  10^  in.  are 
wanted?     If  letter  heads  8^  in.  X  n/4  in-  are  wanted? 

9.  Use  the  size  sheets  that  will  be  more  economical  for 
cutting  half-size  letter  heads  8%  in.  X  $%  in.  and  find  the 
number  of  sheets  you  must  use  to  print  $M  such  letter  heads 
allowing  10%  for  press  waste. 

148 


PRINT    SHOP  149 

10.  An  order  for  2oM  slips  3^  in.  X  2  in.  was  sent  to  your 
print  shop.     How  shall  these  be  set  up  in  order  that  they  may 
be  printed  with  2M  impressions  of  the  press?     What  will  be 
the  dimensions  of  the  sheet  on  which  they  are  printed?     How 
many  sheets  will  be  required  allowing  10%  for  press  waste? 

11.  How  many  sheets  of  cardboard  22  in.  X  28  in.  will  be 
required  for  backs  used  in  padding  the  slips  of  problem  10,  if 
50  slips  are  put  in  each  pad? 

12.  The  cardboard  back  of  a  certain  calendar  is  8%  in.  X 
6%  in.     If  they  were  cut  from  sheets  22  in.  X  28  in.,  how  many 
sheets  were  required  for  10,000  calendars  allowing  10%  for 
waste  in  printing? 

13.  How  many  cards  n  in.  X  14  in.  can  be  cut  from  1000 
sheets  22  in.  X  28  in.     If  the  sheets  cost  $3  per  100,  find  the 
cost  of  the  cards  per  1000,  adding  10%  for  waste  in  printing. 


LESSON  LXXVII 
PRINT  SHOP 

1.  Find  the  number  of  lines  of  i6-point  type  spaced  with  2- 
point  leads  that  can  be  set  in  10  in.;  in  15  in.;  in  12  in.;  in 
53^  in.;  in  24  in. 

2.  The  page  of  a  certain  book  contains  33  lines.     If  the  type 
is  set  solid  and  the  composition  is  33  picas  long  what  is  the  size 
of  the  type  used? 

3.  It  is  required  to  set  solid  72  lines  in  a  i2-in.  space.     What 
size  type  must  be  used?     What  size  type  must  be  used  if  the 
lines  are  spaced  with  2-point  leads? 

4.  Find  the  cost  of  setting  the  page  in  problem  2  at  55^.  per 
M  ems. 

5.  If  65  ems  of  composition  contain  25  words,  how  many 
ems  in  an  article  of  4000  words? 

6.  If  10,000  cards  4  in.  X  6  in.  are  to  be  printed  with  2500 
impressions  of  the  press,  assuming  no  waste,  how  shall  they 
be  set  up?     What  size  sheets  must  be  used?     How  long  will  it 
take  to  print  them  if  the  press  averages  25  impressions  per 
min.? 

7.  How  long  will  it  take  to  print  the  cards  19)^  in.  X  14  in. 
that  can  be  cut  from  2000  sheets  22  in.  X  28  in.  if  the  press 
averages  1200  impressions  per  hr.? 

8.  The  page  of  a  certain  book  averages  34  words  for  each  84 
ems.     How  many  words  in  three  pages  of  this  book  if  the  page 
is  21  picas  X  33  picas? 

9.  Advertising  announcements  83^  in.  X  1 2  in.  are  cut  from 
sheets  22'^  in.  X  28^  in.     Find  the  number  of  sheets  required 
to  print  150  of  the  "ads"  if  10  are  allowed  for  press  waste. 

10.  Find  the  time  required  to  print  the  letter  heads  8  in.  X 

150 


PRINT    SHOP  151 

10^2  in-  that  can  be  cut  from  800  sheets  16  in.  X  21  in.  when 
the  press  averages  18  impressions  per  min. 

11.  A  certain  job  can  be  completed  in  i  hr.  45  min.  if  the 
press  averages  1200  impressions  per  hr.     What  must  be  the 
average  number  of  impressions  per  hour  to  complete  the  job  in 
2  hr.?     In  i  hr.  30  min.? 

12.  A  piece  of  printed  matter  set  solid  16  picas  X  32  picas 
contains  1152  ems.     Find  the  number  of  points  for  each  em. 

13.  A  certain  page  20  picas  X  30  picas  contains  864  ems. 
What  size  type  is  used  for  the  page  if  the  work  is  set  solid? 


LESSON  LXXVIII 
PRINT  SHOP 

1.  Common  type  is  60%  lead,  30%  antimony  and  10%  tin. 
How  much  of  each  in  400  Ib.  of  type?     620  lb.?     157  lb.? 
235  lb.? 

2.  The  best  type  is  50%  lead,  25%  tin  and  25%  antimony. 
How  many  pounds  of  lead  and  of  antimony  must  be  melted 
with  75  lb.  of  tin  to  make  this  grade  of  type? 

3.  In  a  2oo-lb.  font  of  best  type,  how  many  pounds  of  each 
of  the  metals? 

4.  Type  metal  is  77.5%  lead,  6.5%  tin  and  16%  antimony. 
How  many  pounds  of  type  metal  can  be  made  from  64  lb.  of 
antimony?     How  many  pounds  of  lead  and  of  tin  must  be 
used? 

6.  Type  is  also  made  by  melting  5  lb.  of  tin,  9  lb.  of  anti- 
mony, 35  lb.  of  lead  and  i  lb.  of  copper.  Each  metal  is  what 
per  cent,  of  the  alloy? 

6.  A  2-in.  pulley  is  belted  to  a  2o-in.  pulley  on  a*  counter- 
shaft.   The  counter  contains  a  i5-tooth  gear  that  meshes 
with  a  Qo-tooth  gear.     A  revolution  of  the  go-tooth  gear 
makes  one  impression  of  the  press.     If  the  driving  pulley  is 
making  500  R.P.M.,  how  many  impressions  does  the  press 
make?     1000  R.P.M.?     800  R.P.M.?     1250  R.P.M?     850 
R.P.M.? 

7.  In  order  that  the  number  of  impressions  of  the  press  for 
any  given  time  may  be  the  same,  what  change  would  have  to 
be  made  in  the  driven  pulley  if  a  3-in.  pulley  is  substituted 
for  the  2-in.  pulley? 

8.  A  label  for  a  can  6  in.  high  and  6  in.  in  diameter  is 
required.     The  label  is  to  extend  three-fourths  of  the  distance 

152 


PRINT    SHOP  153 

around  the  can  and  within  i  in.  of  the  top  and  %  in.  of  the 
bottom.  How  many  sheets  25  in.  X  38  in.  will  be  required  for 
6000  of  these  labels  allowing  8%  for  waste? 

9.  The  page  of  a  certain  magazine  contains  three  columns 
each  1 8  picas  wide;  between  each  column  is  i  pica.    The 
margin  on  the  right-hand  side  of  the  page  is  5^  picas  and 
there  is  an  equal  margin  at  the  left.     Find  the  width  of  the 
page  in  inches. 

10.  The  printed  matter  of  the  page  of  problem  9  is  set  solid 
and  has  9  lines  per  in.     What  size  type  is  used?     How  many 
8-point  ems  are  there  per  inch? 


LESSON  LXXIX 
FRACTIONAL  REVIEW 
5%  X  iH?  - 


1    simplify 

p  y 


2.  Simplify  ^*-±ia7ji  ~  JTT  ~  ?M 


3.  isH  X  12^3  X  5M  X  6M,=  ?     %  X  7K  X  3%  =  ? 
X  7%  X  8>£  X  25  =  ?     18%  X  9^7  X  3^  =  ? 


4.  Simplify    1X7         .  ,  X  H  of  - 
i>^  of  %  ^  ioH  13 

5    Simplify  M  +  KsXH-KXfr 
mpMy     MXWi+XXM 

6.  What  is  the  value  of  3ff7f*£* 

4/13  Of  2M.6 

7.  Simplify  '-Hrf 


9.  4^8  +  2  x  $%  -  3  x  %  +  y2  =  ? 

10.  %  +  K2  +  KG  +  Ko  -  %  -  Mo 

11.  Find  the  sum  of  ^  +  %  +  %Q  +  Mo  correct  to  four 
decimal  places. 

12.  12^  X  4M  X  5K2  =  ?     1748  +  (6M  X  7^2)  =  ? 

X  i5K2  X  %  =  ?    ^2  X  K2  X  18%  =  ? 

x  ISM  x  $y2  =  ?  1890  -=-  (7y3  x  ioH)  =  ? 

X  9%  X  H  =  ?     ^2  X  sM  X  %  =  ? 

154 


UC  SOUTHERN  REG10IWL  LIBRARY  FACILITY 


A     000933218     0 


ANGELES 
STATE  NORMAL  SCHOOL 


